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Let $$A(p,q) = \sum_{k=1}^{\infty} \frac{(-1)^{k+1}H^{(p)}_k}{k^q},$$ where $H^{(p)}_n = \sum_{i=1}^n i^{-p}$, the $n$th $p$-harmonic number. The $A(p,q)$'s are known as alternating Euler sums.

Can someone provide a nice proof that $$A(1,1) = \sum_{k=1}^{\infty} \frac{(-1)^{k+1} H_k}{k} = \frac{1}{2} \zeta(2) - \frac{1}{2} \log^2 2?$$

I worked for a while on this today but was unsuccessful. Summation by parts, swapping the order of summation, and approximating $H_k$ by $\log k$ were my best ideas, but I could not get any of them to work. (Perhaps someone else can?) I would like a nice proof in order to complete my answer here.

Bonus points for proving $A(1,2) = \frac{5}{8} \zeta(3)$ and $A(2,1) = \zeta(3) - \frac{1}{2}\zeta(2) \log 2$, as those are the other two alternating Euler sums needed to complete my answer.


Added: I'm going to change the accepted answer to robjohn's $A(1,1)$ calculation as a proxy for the three answers he gave here. Notwithstanding the other great answers (especially the currently most-upvoted one, the one I first accepted), robjohn's approach is the one I was originally trying. I am pleased to see that it can be used to do the $A(1,1)$, $A(1,2)$, and $A(2,1)$ derivations.

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  • $\begingroup$ Write out the infinite product $\prod_{k=1}^\infty(1-x^k/k^s)$ in terms of the gamma function, use veitas formulas to gather up partial sums of 1/k^s, and align them with the products corresponding power series, euler did something similar to solve the basel problem. Try doing it for the case s=2, in which case the product is sin(pix)/(pix). $\endgroup$
    – Ethan
    Jan 11, 2013 at 5:34
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    $\begingroup$ @Ethan: Spell out the details, and perhaps that becomes an answer? :) $\endgroup$ Jan 11, 2013 at 5:36
  • $\begingroup$ Can you inculde the defintion of $A(p,q)$ ? $\endgroup$
    – Amr
    Jan 11, 2013 at 6:07
  • $\begingroup$ @Ethan I missed that $\endgroup$
    – Amr
    Jan 11, 2013 at 6:08
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    $\begingroup$ It is interesting that this question has been marked as a favorite $6$ favorites whereas it only has $4$ up-votes. $\endgroup$
    – user17762
    Jan 11, 2013 at 22:08

16 Answers 16

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Note that $$\dfrac{(-1)^{k-1}}k = \int_0^1 (-x)^{k-1}dx$$ and $$\dfrac1n = \int_0^1 y^{n-1}dy$$


For the first one, \begin{align} \sum_{k=1}^{\infty} \dfrac{(-1)^{k+1}}k \sum_{n=1}^k \dfrac1n & = \sum_{k=1}^{\infty} \sum_{n=1}^k \int_0^1 (-x)^{k-1}dx \int_0^1 y^{n-1} dy\\ & = \sum_{n=1}^{\infty} \sum_{k=n}^{\infty} \int_0^1 (-x)^{k-1}dx \int_0^1 y^{n-1} dy\\ & = \sum_{n=1}^{\infty} \int_0^1 \dfrac{(-x)^{n-1}}{1+x}dx \int_0^1 y^{n-1} dy\\ & = \int_0^1 \int_0^1\sum_{n=1}^{\infty} \dfrac{(-xy)^{n-1}}{1+x}dx dy\\ & = \int_0^1 \int_0^1\dfrac1{(1+x)(1+xy)}dx dy\\ & = \int_0^1 \int_0^1\dfrac1{(1+x)(1+xy)}dy dx\\ & = \int_0^1 \dfrac{\log(1+x)}{x(1+x)} dx\\ & = \int_0^1 \dfrac{\log(1+x)}{x} dx - \int_0^1 \dfrac{\log(1+x)}{(1+x)} dx\\ & = \dfrac{\zeta(2)}2 - \dfrac{\log^2 2}2 \end{align}

$$\int_0^1 \dfrac{\log(1+x)}{x} dx = \sum_{k=0}^{\infty} \int_0^1 \dfrac{(-1)^kx^k}{k+1} dx = \sum_{k=0}^{\infty} \dfrac{(-1)^k}{(k+1)^2} = \dfrac{\zeta(2)}2$$ $$\int_0^1 \dfrac{\log(1+x)}{(1+x)} dx = \left. \dfrac{\log^2(1+x)}2 \right \vert_{x=0}^{x=1} = \dfrac{\log^2 2}2$$


For the second one,

$$A(1,2) = \sum_{k=1}^{\infty} \dfrac{(-1)^{k+1}}{k^2} \sum_{n=1}^k \dfrac1n $$ $$\dfrac{(-1)^{k-1}}{k^2} = \int_0^1 (-x)^{k-1} dx \int_0^1 z^{k-1} dz = (-1)^{k-1} \int_0^1 \int_0^1 (xz)^{k-1} dx dz$$ \begin{align} \sum_{k=1}^{\infty} \dfrac{(-1)^{k+1}}{k^2} \sum_{n=1}^k \dfrac1n & = \sum_{k=1}^{\infty} \sum_{n=1}^k \int_0^1\int_0^1 (-1)^{k-1} (xz)^{k-1}dxdz \int_0^1 y^{n-1} dy\\ & = \int_0^1 \int_0^1 \int_0^1 \sum_{n=1}^{\infty} \dfrac{(-xyz)^{n-1}}{1+xz} dx dy dz\\ & = \int_0^1 \int_0^1 \int_0^1 \dfrac1{(1+xz)(1+xyz)} dx dy dz\\ & = \int_0^1 \int_0^1 \dfrac{\log(1+xz)}{xz(1+xz)} dx dz\\ & = \int_0^1 \int_0^1 \dfrac{\log(1+xz)}{xz} dx dz - \int_0^1 \int_0^1 \dfrac{\log(1+xz)}{1+xz} dx dz\\ & = \int_0^1 \int_0^1 \dfrac{\log(1+xz)}{xz} dx dz- \int_0^1 \dfrac{\log^2(1+z)}{2z} dz\\ & = \dfrac34 \zeta(3) - \dfrac{\zeta(3)}8\\ & = \dfrac58 \zeta(3) \end{align}

$$ \int_0^1 \int_0^1 \dfrac{\log(1+xz)}{xz} dx dz = \sum_{k=0}^{\infty} \int_0^1 \int_0^1 \dfrac{(-1)^k (xz)^k}{k+1} dx dz = \sum_{k=0}^{\infty} \dfrac{(-1)^k}{(k+1)^3} = \dfrac34 \zeta(3)$$


For the third one, $$A(2,1) = \sum_{k=1}^{\infty} \dfrac{(-1)^{k+1}}{k} \sum_{n=1}^k \dfrac1{n^2} $$ \begin{align} \sum_{k=1}^{\infty} \dfrac{(-1)^{k+1}}{k} \sum_{n=1}^k \dfrac1{n^2} & = \int_0^1 \int_0^1 \int_0^1 \sum_{k=1}^{\infty} \sum_{n=1}^k (-1)^{k-1} x^{k-1} (yz)^{n-1} dx dy dz\\ & = \int_0^1 \int_0^1 \int_0^1 \sum_{n=1}^{\infty} \sum_{k=n}^{\infty} (-1)^{k-1} x^{k-1} (yz)^{n-1} dx dy dz\\ & = \int_0^1 \int_0^1 \int_0^1 \sum_{n=1}^{\infty} \dfrac{(-xyz)^{n-1}}{1+x} dx dy dz\\ & = \int_0^1 \int_0^1 \int_0^1 \dfrac1{(1+x)(1+xyz)} dx dy dz\\ & = \int_0^1 \int_0^1 \dfrac{\log(1+xy)}{(1+x)(xy)} dx dy\\ & = \zeta(3) - \dfrac{\zeta(2) \log 2}2 \end{align}


In general, if I have not made any mistake, this can be extended to $A(p,q)$. $$A(p,q) = \underbrace{\int_0^1 \int_0^1 \cdots \int_0^1}_{p+q \text{ times}} \dfrac{dx_1 dx_2 \cdots dx_{p+q}}{(1+x_1 x_2 \cdots x_q)(1+x_1 x_2 \cdots x_{p+q})}$$


Proceeding along similar lines, we also get that $$B(p,q) = \sum_{k=1}^{\infty} \dfrac{H_k^{(p)}}{k^q} = \underbrace{\int_0^1 \int_0^1 \cdots \int_0^1}_{p+q \text{ times}} \dfrac{dx_1 dx_2 \cdots dx_{p+q}}{(1-x_1 x_2 \cdots x_q)(1-x_1 x_2 \cdots x_{p+q})}$$


We also get that $$C(p,q) = \sum_{k=1}^{\infty} \dfrac1{k^q} \sum_{i=1}^k \dfrac{(-1)^{i-1}}{i^p} = \underbrace{\int_0^1 \int_0^1 \cdots \int_0^1}_{p+q \text{ times}} \dfrac{dx_1 dx_2 \cdots dx_{p+q}}{(1-x_1 x_2 \cdots x_q)(1+x_1 x_2 \cdots x_{p+q})}$$ $$D(p,q) = \sum_{k=1}^{\infty} \dfrac{(-1)^{k-1}}{k^q} \sum_{i=1}^k \dfrac{(-1)^{i-1}}{i^p} = \underbrace{\int_0^1 \int_0^1 \cdots \int_0^1}_{p+q \text{ times}} \dfrac{dx_1 dx_2 \cdots dx_{p+q}}{(1+x_1 x_2 \cdots x_q)(1-x_1 x_2 \cdots x_{p+q})}$$


By the same argument as above, in general, nested sums like $$\sum_{k=1}^{\infty} \dfrac{(\pm 1)^{k-1}}{k^q} \sum_{n=1}^k \dfrac{(\pm 1)^{n-1}}{n^p} \sum_{m=1}^n \dfrac{(\pm 1)^{m-1}}{m^r} \cdots $$ equals $$\underbrace{\int_0^1 \int_0^1 \cdots \int_0^1}_{p+q+r+\cdots \text{ times}} \dfrac{dx_1 dx_2 \cdots dx_{p+q+r+\cdots}}{(1\mp x_1 \cdots x_q)(1(\mp)(\pm)x_1 \cdots x_{p+q}) \cdots (1(\mp)(\pm)\cdots(\pm)x_1 \cdots x_{p+q+r+\cdots})}$$

For instance, $$\sum_{k=1}^{\infty} \dfrac{1}{k^q} \sum_{n=1}^k \dfrac{1}{n^p} \sum_{m=1}^n \dfrac{1}{m^r} = \underbrace{\int_0^1 \int_0^1 \cdots \int_0^1}_{p+q+r \text{ times}} \dfrac{dx_1 dx_2 \cdots dx_{p+q+r}}{(1- x_1 \cdots x_q)(1-x_1 \cdots x_{p+q}) \cdots (1-x_1 \cdots x_{p+q+r})}$$ $$\sum_{k=1}^{\infty} \dfrac{(-1)^{k-1}}{k^q} \sum_{n=1}^k \dfrac{1}{n^p} \sum_{m=1}^n \dfrac{1}{m^r} = \underbrace{\int_0^1 \cdots \int_0^1}_{p+q+r \text{ times}} \dfrac{dx_1 dx_2 \cdots dx_{p+q+r}}{(1+ x_1 \cdots x_q)(1+x_1 \cdots x_{p+q}) \cdots (1+x_1 \cdots x_{p+q+r})}$$ $$\sum_{k=1}^{\infty} \dfrac{(-1)^{k-1}}{k^q} \sum_{n=1}^k \dfrac{(-1)^{n-1}}{n^p} \sum_{m=1}^n \dfrac{1}{m^r} = \underbrace{\int_0^1 \cdots \int_0^1}_{p+q+r \text{ times}} \dfrac{dx_1 dx_2 \cdots dx_{p+q+r}}{(1+ x_1 \cdots x_q)(1-x_1 \cdots x_{p+q}) \cdots (1-x_1 \cdots x_{p+q+r})}$$ $$\sum_{k=1}^{\infty} \dfrac{1}{k^q} \sum_{n=1}^k \dfrac{(-1)^{n-1}}{n^p} \sum_{m=1}^n \dfrac{1}{m^r} = \underbrace{\int_0^1 \cdots \int_0^1}_{p+q+r \text{ times}} \dfrac{dx_1 dx_2 \cdots dx_{p+q+r}}{(1- x_1 \cdots x_q)(1+x_1 \cdots x_{p+q}) \cdots (1+x_1 \cdots x_{p+q+r})}$$


Similarly, for negative $p$,$q$ $r$ etc, we can replace the integrals $\int_0^1$ by the appropriate differentiation operator evaluated at $1$. I will post this in detail sometime over the weekend.

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    $\begingroup$ @Marvis: a powerful answer. We need such answers on MSE. (+1) $\endgroup$ Jan 11, 2013 at 7:21
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    $\begingroup$ I am completely astounded by this brilliant answer! I also succeeded in calculating those series by using dilogarithm identities and trilogarithm identity, my solutions lack a systematic approach that your illuminating answer shows. That's why I love this answer. $\endgroup$ Jan 11, 2013 at 18:11
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    $\begingroup$ This deserves a place in american mathematical monthly $\endgroup$
    – Norbert
    Jan 12, 2013 at 23:14
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    $\begingroup$ I feel completely overwhelmed solely by skimming this answer, Marvis. Dare I call it beautiful, for I know not what it is . . . $\endgroup$
    – 000
    Jan 13, 2013 at 3:22
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    $\begingroup$ My answer seems like a dinghy next to your aircraft carrier :-) $\endgroup$
    – robjohn
    Sep 20, 2013 at 16:04
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$A(1,1)$: $$ \begin{align} \sum_{n=1}^N\frac{(-1)^{n-1}}{n}H_n &=\sum_{n=1}^N\frac{(-1)^{n-1}}{n^2}+\sum_{n=2}^N\frac{(-1)^{n-1}}{n}H_{n-1}\\ &=\sum_{n=1}^N\frac{(-1)^{n-1}}{n^2}+\frac12\sum_{n=2}^N\sum_{k=1}^{n-1}\frac{(-1)^{n-1}}{n}\left(\frac1k+\frac1{n-k}\right)\\ &=\sum_{n=1}^N\frac{(-1)^{n-1}}{n^2}+\frac12\sum_{n=2}^N\sum_{k=1}^{n-1}\frac{(-1)^{n-1}}{k(n-k)}\\ &=\sum_{n=1}^N\frac{(-1)^{n-1}}{n^2}+\frac12\sum_{k=1}^{N-1}\sum_{n=k+1}^N\frac{(-1)^{n-1}}{k(n-k)}\\ &=\sum_{n=1}^N\frac{(-1)^{n-1}}{n^2}+\frac12\sum_{k=1}^{N-1}\sum_{n=1}^{N-k}\frac{(-1)^{n+k-1}}{kn}\\ &=\color{#00A000}{\sum_{n=1}^N\frac{(-1)^{n-1}}{n^2}} -\color{#0000FF}{\frac12\sum_{k=1}^{N-1}\frac{(-1)^{k-1}}{k}\sum_{n=1}^{N-1}\frac{(-1)^{n-1}}{n}}\\ &+\color{#C00000}{\frac12\sum_{k=1}^{N-1}\frac{(-1)^{k-1}}{k}\sum_{n=N-k+1}^{N-1}\frac{(-1)^{n-1}}{n}}\tag{1} \end{align} $$ where, using the Alternating Series Test, we have $$ \begin{align} &\color{#C00000}{\frac12\left|\sum_{k=1}^{N-1}\frac{(-1)^{k-1}}{k}\sum_{n=N-k+1}^{N-1}\frac{(-1)^{n-1}}{n}\right|}\\ &\le\frac12\left|\sum_{k=1}^{N/2}\frac{(-1)^{k-1}}{k}\sum_{n=N-k+1}^{N-1}\frac{(-1)^{n-1}}{n}\right| +\frac12\left|\sum_{k=N/2}^{N-1}\frac{(-1)^{k-1}}{k}\sum_{n=N-k+1}^{N-1}\frac{(-1)^{n-1}}{n}\right|\\ &\le\frac12\cdot1\cdot\frac2N+\frac12\cdot\frac2N\cdot1\\ &=\frac2N\tag{2} \end{align} $$ Applying $(2)$ to $(1)$ and letting $N\to\infty$, we get $$ \sum_{n=1}^\infty\frac{(-1)^{n-1}}{n}H_n=\color{#00A000}{\frac12\zeta(2)}-\color{#0000FF}{\frac12\log(2)^2}\tag{3} $$

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  • $\begingroup$ Very nice answer! I hope it will be highly upvoted! $\endgroup$ Sep 20, 2013 at 15:48
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    $\begingroup$ @Chris'ssis: thanks. I'm not holding my breath since I came across this question so late. $\endgroup$
    – robjohn
    Sep 20, 2013 at 15:50
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    $\begingroup$ very good answer (+1) . $\endgroup$
    – what'sup
    Sep 20, 2013 at 18:18
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    $\begingroup$ Rob, this is very nice! I particularly like the fact that it is the kind of approach I was originally trying but couldn't get to work. For others reading this, what Rob has done is to extract the eta function partial sum and then show that the rest turns into the square of the partial sum of the alternating harmonic series plus change. It's really clear how the answer falls out this way. Beautiful. $\endgroup$ Sep 20, 2013 at 21:29
  • $\begingroup$ @MikeSpivey: Thanks for the comments and corrections! Let me know if you find anything that needs explanation for my derivation of $A(1,2)$. $\endgroup$
    – robjohn
    Sep 20, 2013 at 21:43
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$A(1,2)$: $$ \begin{align} \sum_{n=1}^\infty\frac1{n^2}H_n &=\sum_{n=1}^\infty\sum_{k=1}^\infty\frac1{n^2}\left(\frac1k-\frac1{k+n}\right)\\ &=\sum_{n=1}^\infty\sum_{k=1}^\infty\frac1{nk(k+n)}\tag{1}\\ &=\sum_{k=1}^\infty\sum_{n=k+1}^\infty\frac1{nk(n-k)}\\ &=\sum_{n=2}^\infty\sum_{k=1}^{n-1}\frac1{nk(n-k)}\\ &=\sum_{n=2}^\infty\sum_{k=1}^{n-1}\frac1{n^2}\left(\frac1k+\frac1{n-k}\right)\\ &=2\sum_{n=1}^\infty\frac1{n^2}H_{n-1}\\ &=2\sum_{n=1}^\infty\frac1{n^2}H_n-2\zeta(3)\tag{2}\\ \sum_{n=1}^\infty\frac1{n^2}H_n &=2\zeta(3)\tag{3} \end{align} $$ $$ \begin{align} \sum_{n=1}^\infty\frac{(-1)^n}{n^2}H_n &=\sum_{n=1}^\infty\sum_{k=1}^\infty\frac{(-1)^n}{n^2}\left(\frac1k-\frac1{k+n}\right)\\ &=\sum_{n=1}^\infty\sum_{k=1}^\infty\frac{(-1)^n}{nk(k+n)}\tag{4}\\ \sum_{n=1}^\infty\frac{(-1)^n}{n^2}H_n &=-\frac34\zeta(3)+\sum_{n=1}^\infty\frac{(-1)^n}{n^2}H_{n-1}\\ &=-\frac34\zeta(3)+\frac12\sum_{n=1}^\infty\sum_{k=1}^{n-1}\frac{(-1)^n}{n^2}\left(\frac1k+\frac1{n-k}\right)\\ &=-\frac34\zeta(3)+\frac12\sum_{k=1}^\infty\sum_{n=k+1}^\infty\frac{(-1)^n}{nk(n-k)}\\ &=-\frac34\zeta(3)+\frac12\sum_{k=1}^\infty\sum_{n=1}^\infty\frac{(-1)^{n+k}}{(n+k)kn}\tag{5} \end{align} $$ Using $\color{#C00000}{(1)}$, $\color{#C00000}{(3)}$, $\color{#00A000}{(4)}$, $\color{#0000FF}{(4)}$, and $\color{#C0A000}{(5)}$ along with the fact that $1+(-1)^k+(-1)^n+(-1)^{n+k}=4$ iff $k$ and $n$ are both even and $0$ otherwise: $$ \begin{align} \zeta(3) &=\frac12\sum_{k=1}^\infty\sum_{n=1}^\infty\frac1{nk(n+k)}\\ &=\sum_{k=1}^\infty\sum_{n=1}^\infty\frac{\color{#C00000}{1}+\color{#00A000}{(-1)^k}+\color{#0000FF}{(-1)^n}+\color{#C0A000}{(-1)^{n+k}}}{nk(n+k)}\\ &=\color{#C00000}{2\zeta(3)}+\color{#00A000}{\sum_{n=1}^\infty\frac{(-1)^n}{n^2}H_n}+\color{#0000FF}{\sum_{n=1}^\infty\frac{(-1)^n}{n^2}H_n} +\color{#C0A000}{2\sum_{n=1}^\infty\frac{(-1)^n}{n^2}H_n+\frac32\zeta(3)}\\ \hspace{-8mm}-\frac58\zeta(3) &=\sum_{n=1}^\infty\frac{(-1)^n}{n^2}H_n\tag{6} \end{align} $$ That is, $$ \sum_{n=1}^\infty\frac{(-1)^{n-1}}{n^2}H_n=\frac58\zeta(3)\tag{7} $$

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    $\begingroup$ An amazing answer! (+1) If I could I'd upvote it $1000$ times. $\endgroup$ Sep 20, 2013 at 16:37
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    $\begingroup$ @Chris'ssis: I spent pretty much all day yesterday on this sum for a question you asked in chat, then saw that it was asked in this question, too! :-) $\endgroup$
    – robjohn
    Sep 20, 2013 at 16:42
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    $\begingroup$ an amazing answer (+1) $\endgroup$
    – what'sup
    Sep 20, 2013 at 18:19
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    $\begingroup$ Again, very nice! It would never have occurred to me to use the property that $1 + (-1)^k + (-1)^n + (-1)^{n+k} = 4$ when $n$ and $k$ are even and vanishes otherwise. Well done. $\endgroup$ Sep 20, 2013 at 21:53
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    $\begingroup$ @Jeff: See this answer with $q=2$, then note that $$\sum_{n\text { odd}}a_n=\frac12\left(\sum_{n=1}^\infty a_n+\sum_{n=1}^\infty(-1)^{n-1}a_n\right)$$ $\endgroup$
    – robjohn
    Mar 1, 2014 at 20:33
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Using integral representation: $$ A(1,1)= \sum_{n=1}^\infty \frac{(-1)^{n+1}}{n} H_n = -\int_0^1 \sum_{n=1}^\infty (-x)^n H_n \frac{\mathrm{d} x }{x} $$ Now: $$ -\sum_{n=1}^\infty (-x)^n H_n = -\sum_{n=1}^\infty x^n \sum_{k=0}^{n-1} (-1)^k \frac{(-1)^{n-k}}{n-k} = -\sum_{n=0}^\infty (-x)^n \cdot \sum_{k=1}^\infty \frac{(-x)^k}{k} = \frac{\log(1+x)}{1+x} $$ Thus $$ A(1,1) = \int_0^1 \frac{\log(1+x)}{1+x} \frac{\mathrm{d}x}{x} = \left. \left(-\frac{1}{2} \log^2(1+x) - \operatorname{Li}_2(-x) \right)\right|_{x = 0}^{x=1} = -\frac{1}{2} \log^2(2) - \operatorname{Li}_2(-1) $$ But $\operatorname{Li}_2(-1) = \sum_{k=1}^\infty \frac{(-1)^k}{k^2} = \left(2^{1-2}-1\right) \zeta(2) = -\frac{1}{2} \zeta(2)$. Thus$$ A(1,1) = \frac{1}{2} \left( \zeta(2) - \log^2(2)\right) $$

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  • $\begingroup$ Thanks, Sasha. One question, though: I don't follow the step $-\sum_{n=1}^\infty x^n \sum_{k=0}^{n-1} (-1)^k \frac{(-1)^{n-k}}{n-k} = -\sum_{n=0}^\infty (-x)^n \cdot \sum_{k=1}^\infty \frac{(-x)^k}{k}$. $\endgroup$ Jan 11, 2013 at 17:47
  • $\begingroup$ Never mind; I got it. You're using the product formula for power series. Thanks again. $\endgroup$ Jan 11, 2013 at 17:50
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Related problems: (I), (II), (III), (IV), $(5)$. For $A(1, 1)$, one can have the integral representation

$$ A(1,1) = \int _{1}^{2}\!{\frac {\ln \left( t \right) }{t \left( t-1 \right) }} {dw}.$$

In general, one can have the following representation for $A(p,1)$

$$ A(p,1) = -\int _{0}^{1}\!{\frac { Li_{p}\left( -u \right) }{ \left( 1+ u \right) u}}{du},$$

where $Li_{p}(-u)$ is the polylogarithm function. Here are some numerical values for $p$ from $1$ to $5$

$$ 0.5822405265,\, 0.6319661978,\, 0.6603570751,\, 0.6759332433,\, 0.6842426955. $$

The General Case A(p,q):

$$ A(p,q) =\sum_{k=1}^{\infty} \frac{(-1)^{k+1}H^{(p)}_k}{k^q} = \frac{\left( -1 \right) ^{q}}{\Gamma(q)}\int _{0}^{1}\!{\frac { \left( \ln\left( u \right) \right)^{q-1}{Li_{p}(-u)} }{ u\left( 1+ u \right) }}{du}. $$

Some numerical values

$$ A(1,2) = .7512855645,\, A(2, 3) = .8793713030, \, A(3, 4) = .9407280160, $$

$$ A(2,1) = .6319661978, A(3, 2) = .8024944234, A(4, 3) = .8956823180. $$

Added

The General Case B(p,q):

$$ B(p,q) = \sum_{k=1}^{\infty} \dfrac{H_k^{(p)}}{k^q}=\frac{(-1)^q}{\Gamma(q)}\int_{0}^{1}\!{\frac {\left(\ln\left(u\right)\right)^{q-1}{Li_{p}(u)} }{ u\left( u-1 \right)}}{du}. $$

Some numerical values

$$ B(1, 2) = 2.404113806, B(2, 3) = 1.265738152, B(3, 4) = 1.093509100, $$

$$ B(3, 2) = 1.748493953, B(4, 3) = 1.215854292, B(5, 4) = 1.084986223. $$

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  • $\begingroup$ Thanks, but that's already in part of Marvis's answer. $\endgroup$ Jan 12, 2013 at 23:04
  • $\begingroup$ @MikeSpivey: You are welcome. See the more general case $A(p,1)$. $\endgroup$ Jan 12, 2013 at 23:35
  • $\begingroup$ Your generalization to the $A(p,1)$ case is interesting. Do you have a proof or reference for that? $\endgroup$ Jan 13, 2013 at 3:09
  • $\begingroup$ A proof for $A(p,1)$ is as follows. $$A(p,1) = \underbrace{\int_0^1 \cdots \int_0^1}_{p+1 \text{ times}} \int_0^1 \dfrac{dx_1 dx_2 \cdots dx_{p+1}}{(1+x_1)(1+x_1x_2x_3 \cdots x_{p+1})}$$ Now $$\underbrace{\int_0^1 \cdots \int_0^1}_{p \text{ times}} \int_0^1 \dfrac{dx_2 dx_3 \cdots dx_{p+1}}{(1+x_1x_2x_3 \cdots x_{p})} = -\dfrac{\text{Li}_p(-x_1)}{x_1}$$ which follows immediately from induction and definition of $\text{Li}_n(x)$. $\endgroup$
    – user17762
    Jan 13, 2013 at 3:20
  • $\begingroup$ @Marvis: Thanks. +1 to Mhenni's answer. $\endgroup$ Jan 13, 2013 at 3:28
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Actually it suffices to know the generating function

$$\sum_{k\geq 1}H^{(p)}_kx^k=\frac{\mathrm{Li}_p(x)}{1-x}$$

Upon integrating we obtain

$$\sum_{k\geq 1}\frac{H^{(p)}_k}{k}x^k=\mathrm{Li}_{p+1}(x)+\int^x_0 \frac{\mathrm{Li}_p(t)}{1-t}\,d t$$

$$\sum_{k\geq 1}\frac{H_k}{k}x^k=\mathrm{Li}_{2}(x)+\frac{1}{2}\log^2(1-x)$$

$$\sum_{k\geq 1}\frac{H_k}{k}(-1)^k=-\frac{\pi^2}{12}+\frac{1}{2}\log^2(2)$$

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    $\begingroup$ Typo on the last line. Should be $\sum_{k\geq 1}\frac{H_k}{k}(-1)^k$, I believe. $\endgroup$ Aug 24, 2014 at 2:45
  • $\begingroup$ @columbus8myhw, thanks. $\endgroup$ Oct 22, 2014 at 4:16
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$A(2,1)$:

$$ \begin{align} \sum_{n=1}^\infty(-1)^{n-1}\frac{H_n^{(2)}}{n} &=\sum_{n=1}^\infty\frac{(-1)^{n-1}}{n^3}+\sum_{n=1}^\infty(-1)^{n-1}\frac{H_{n-1}^{(2)}}{n}\tag{1}\\ &=\frac34\zeta(3)+\sum_{n=1}^\infty\frac{(-1)^{n-1}}{n}\sum_{k=1}^{n-1}\frac1{k^2}\tag{2}\\ &=\frac34\zeta(3)+\sum_{k=1}^\infty\sum_{n=k+1}^\infty\frac{(-1)^{n-1}}{nk^2}\tag{3}\\ &=\frac34\zeta(3)+\sum_{k=1}^\infty\sum_{n=1}^\infty\frac{(-1)^{k+n-1}}{(k+n)k^2}\tag{4}\\ &=\frac34\zeta(3)+\sum_{k=1}^\infty\sum_{n=1}^\infty(-1)^{k+n-1}\left(\frac1{k^2n}-\frac1{kn(k+n)}\right)\tag{5}\\[6pt] &=\frac34\zeta(3)-\frac12\zeta(2)\log(2)+\frac14\zeta(3)\tag{6}\\[9pt] &=\zeta(3)-\frac12\zeta(2)\log(2)\tag{7} \end{align} $$ Justification:
$(1)$: $H_n^{(2)}=\frac1{n^3}+H_{n-1}^{(2)}$
$(2)$: expand $H_{n-1}^{(2)}$
$(3)$: change order of summation
$(4)$: reindex $n\mapsto k+n$
$(5)$: $\frac1{(k+n)k^2}=\frac1{k^2n}-\frac1{kn(k+n)}$
$(6)$: $\sum\limits_{k=1}^\infty\sum\limits_{n=1}^\infty\frac{(-1)^{k+n}}{kn(k+n)}=\frac14\zeta(3)$ from $(5)$ and $(7)$ of this answer
$(7)$: addition

Note that this answer was taken from this answer. There, it is shown, using the Euler Series Transformation, that $$ A(2,1)=\sum_{n=1}^\infty\frac{H_n}{2^nn^2}\tag{8} $$

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  • $\begingroup$ Thanks, Rob. I appreciate you taking the time to add this answer. $\endgroup$ Dec 17, 2013 at 19:34
  • $\begingroup$ @robjohn I always enjoy reading your posts, not only for their creative content, but also the style you've adopted to facilitate the reader's understanding of the development. And I've notice that you often embed references to previously answered questions. Rob, this might not be the right place to ask, but how do you keep track of your posted answers? I've posted now a bit over 1,000 answers and find it challenging to find the one or ones that I can use as a reference when replying to a new question. Is their a "search" method that you use or have you developed a some catalogue? $\endgroup$
    – Mark Viola
    Sep 17, 2015 at 17:28
  • $\begingroup$ @Dr.MV: I used to try to keep track of the answers that I thought I might reference by marking the questions as favorites. However, that started getting crowded, so I keep a number of answers in an offline catalogue. That allows for better organization and easier searching. $\endgroup$
    – robjohn
    Sep 17, 2015 at 21:13
  • $\begingroup$ @robjohn Thanks Rob. I'd like to know more about how you organize, but that is likely asking too much. I'll see if I can implement a scheme that works well. $\endgroup$
    – Mark Viola
    Sep 17, 2015 at 21:30
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Interestingly, $$ \sum_{n=1}^{\infty} \frac{(-1)^{n-1}H_{n}^{-}}{n} = \frac{\zeta(2)}{2} {\color{red}{+}} \frac{\log^{2} (2)}{2}$$ where $H_{n}^{-}$ are the alternating harmonic numbers defined as $$H_{n}^{-} = \sum_{k=1}^{n} \frac{(-1)^{k-1}}{k} .$$

One way to show this is to notice that $$ \begin{align} \log (2) - H_{n}^{-} &= \sum_{k=n+1}^{\infty} \frac{(-1)^{k-1}}{k} \\ &= (-1)^{n}\sum_{k=1}^{\infty} \frac{(-1)^{k-1}}{k+n} \\ &= (-1)^{n} \sum_{k=1}^{\infty} (-1)^{k-1} \int_{0}^{1} x^{k+n-1} \ dx \\ &= (-1)^{n} \int_{0}^{1} x^{n}\sum_{k=1}^{\infty}(-1)^{k-1} x^{k-1} \ dx \\ &= (-1)^{n} \int_{0}^{1} \frac{x^{n}}{1+x} \ dx . \end{align}$$

Thus an integral representation of the alternating harmonic numbers is $$ H_{n}^{-} = \log (2) + (-1)^{n-1} \int_{0}^{1} \frac{x^{n}}{1+x} \ dx .$$

The integral on the right can be evaluated in terms of the digamma function, and you'll get a closed-form expression for the alternating harmonic numbers.

But getting back to evaluating that sum,

$$ \begin{align} \sum_{n=1}^{\infty} \frac{(-1)^{n-1}H_{n}^{-}}{n} &= \log(2) \sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n} + \sum_{n=1}^{\infty} \frac{1}{n} \int_{0}^{1} \frac{x^{n}}{1+x} \ dx \\ &= \log^{2} (2) + \int_{0}^{1} \frac{1}{1+x} \sum_{n=1}^{\infty} \frac{x^{n}}{n} \ dx \\ &= \log^{2} (2) - \int_{0}^{1} \frac{\log (1-x)}{1+x} \ dx \\ &=\log^{2} 2 - \int_{1/2}^{1} \frac{\log \big(1-(2t-1) \big)}{2t} \ 2 \ dt \\ &= \log^{2}(2) - \int_{1/2}^{1} \frac{\log \big(2(1-t) \big)}{t} \ dt \\ &= \log^{2}(2) - \int_{1/2}^{1} \frac{\log 2}{t} \ dt - \int_{1/2}^{1} \frac{\log (1-t)}{t} \ dt \\ &= \log^{2}(2) - \log^{2}(2) + \text{Li}_{2}(1) - \text{Li}_{2} \left( \frac{1}{2}\right) \\ &= \zeta(2) - \frac{\zeta(2)}{2} + \frac{\log^{2} (2)}{2} \\ &= \frac{\zeta (2)}{2} + \frac{\log^{2} (2)}{2} . \end{align}$$

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  • $\begingroup$ That is interesting. Thanks for the observation. $\endgroup$ Jul 24, 2014 at 21:40
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$\newcommand{\+}{^{\dagger}} \newcommand{\angles}[1]{\left\langle\, #1 \,\right\rangle} \newcommand{\braces}[1]{\left\lbrace\, #1 \,\right\rbrace} \newcommand{\bracks}[1]{\left\lbrack\, #1 \,\right\rbrack} \newcommand{\ceil}[1]{\,\left\lceil\, #1 \,\right\rceil\,} \newcommand{\dd}{{\rm d}} \newcommand{\down}{\downarrow} \newcommand{\ds}[1]{\displaystyle{#1}} \newcommand{\expo}[1]{\,{\rm e}^{#1}\,} \newcommand{\fermi}{\,{\rm f}} \newcommand{\floor}[1]{\,\left\lfloor #1 \right\rfloor\,} \newcommand{\half}{{1 \over 2}} \newcommand{\ic}{{\rm i}} \newcommand{\iff}{\Longleftrightarrow} \newcommand{\imp}{\Longrightarrow} \newcommand{\isdiv}{\,\left.\right\vert\,} \newcommand{\ket}[1]{\left\vert #1\right\rangle} \newcommand{\ol}[1]{\overline{#1}} \newcommand{\pars}[1]{\left(\, #1 \,\right)} \newcommand{\partiald}[3][]{\frac{\partial^{#1} #2}{\partial #3^{#1}}} \newcommand{\pp}{{\cal P}} \newcommand{\root}[2][]{\,\sqrt[#1]{\vphantom{\large A}\,#2\,}\,} \newcommand{\sech}{\,{\rm sech}} \newcommand{\sgn}{\,{\rm sgn}} \newcommand{\totald}[3][]{\frac{{\rm d}^{#1} #2}{{\rm d} #3^{#1}}} \newcommand{\ul}[1]{\underline{#1}} \newcommand{\verts}[1]{\left\vert\, #1 \,\right\vert} \newcommand{\wt}[1]{\widetilde{#1}}$ $\ds{\sum_{k = 1}^{\infty}{\pars{-1}^{k + 1} H_{\rm k} \over k}:\ {\large ?}}$

\begin{align}&\color{#c00000}{% \sum_{k = 1}^{\infty}{\pars{-1}^{k + 1} H_{\rm k}\over k}} =\sum_{k = 1}^{\infty}{\pars{-1}^{k + 1} \over k} \int_{0}^{1}{1 - t^{k} \over 1 - t}\,\dd t \\[3mm]&=\sum_{k = 1}^{\infty}{\pars{-1}^{k + 1} \over k}\int_{0}^{1} \ln\pars{1 - t}\pars{-kt^{k - 1}}\,\dd t =-\int_{0}^{1}\ln\pars{1 - t}\sum_{k = 1}^{\infty}\pars{-t}^{k - 1}\,\dd t \\[3mm]&=-\int_{0}^{1}{\ln\pars{1 - t} \over 1 + t}\,\dd t =-\,\int_{0}^{1}{\ln\pars{t} \over 2 - t}\,\dd t =-\,\int_{0}^{1/2}{\ln\pars{2t} \over 1 - t}\,\dd t =-\,\int_{0}^{1/2}{\ln\pars{1 - t} \over t}\,\dd t \\[3mm]&=\int_{0}^{1/2}{{\rm Li}_{1}\pars{t} \over t}\,\dd t \end{align} where $\ds{{\rm Li}_{s}\pars{z}}$ is a PolyLogarithm Function and we'll use well known properties of them as explained in the above mentioned link.

Then, $$ \color{#c00000}{% \sum_{k = 1}^{\infty}{\pars{-1}^{k + 1} H_{\rm k}\over k}} =\int_{0}^{1/2}{\rm Li}_{2}'\pars{t}\,\dd t ={\rm Li}_{2}\pars{\half} - {\rm Li}_{2}\pars{0} =\color{#c00000}{{\rm Li}_{2}\pars{\half}} $$

$\ds{{\rm Li}_{2}\pars{\half}}$ is given in the above mentioned link: \begin{align}&\color{#66f}{\large% \sum_{k = 1}^{\infty}{\pars{-1}^{k + 1} H_{\rm k}\over k}} ={\pi^{2} \over 12} - \half\,\ln^{2}\pars{2} =\color{#66f}{\large\half\bracks{\zeta\pars{2} - \ln^{2}\pars{2}}} \end{align}

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  • $\begingroup$ How do you go from $\int_0^1 \frac{1-t^k}{1-t} \, dt$ to $\int_0^1 \ln(1-t) (-kt^{k-1}) \, dt$ in the second step? $\endgroup$ Jun 9, 2014 at 18:43
  • $\begingroup$ @MikeSpivey Integrating by parts since $\displaystyle\large{{\rm d}\ln\left(1 - t\right) \over {\rm d}t} = -\,{1 \over 1 - t}$. Thanks. $\endgroup$ Jun 9, 2014 at 18:46
  • $\begingroup$ Got it; thanks! And +1. $\endgroup$ Jun 9, 2014 at 18:49
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$$\sum_{k=1}^\infty\frac{(-1)^{k+1}}{k}H_k=\sum_{k=1}^\infty (-1)^{k+1}H_k\int_0^1 x^{k-1}dx\\=\int_0^1\frac1x\sum_{k=1}^\infty{-H_k (-x)^{k}}dx=\int_0^1\frac{\ln(1+x)}{x(1+x)}dx\\=\int_0^1\frac{\ln(1+x)}{x}dx-\int_0^1\frac{\ln(1+x)}{1+x}dx\\=-\operatorname{Li}_2(-1)-\frac12\ln^22\\=\frac12\zeta(2)-\frac12\ln^22$$

where we used the identity $\sum_{n=1}^\infty H_nx^n=-\frac{\ln(1-x)}{1-x}$ and the value $\operatorname{Li}_2(-1)=-\frac12\zeta(2)$

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A full derivation of $A(m,1), \ m\ge2$, is found in this answer, \begin{equation*} \sum_{n=1}^{\infty} (-1)^{n-1}\frac{H_n^{(m)}}{n}=\frac{(-1)^m}{(m-1)!}\int_0^1\frac{\displaystyle \log^{m-1}(x)\log\left(\frac{1+x}{2}\right)}{1-x}\textrm{d}x \end{equation*} \begin{equation*} =\frac{1}{2}\biggr(m\zeta (m+1)-2\log (2) \left(1-2^{1-m}\right) \zeta (m)-\sum_{k=1}^{m-2} \left(1-2^{-k}\right)\left(1-2^{1+k-m}\right)\zeta (k+1)\zeta (m-k)\biggr), \end{equation*} where $H_n^{(m)}=1+\frac{1}{2^m}+\cdots+\frac{1}{n^m}$ represents the $n$th generalized harmonic number of order $m$ and $\zeta$ denotes the Riemann zeta function.

Also, a full solution to the case

\begin{equation*} \sum_{k=1}^{\infty} (-1)^{k-1} \frac{H_k}{k^{2n}}=\left(n+\frac{1}{2}\right)\eta(2n+1)-\frac{1}{2}\zeta(2n+1)-\sum_{k=1}^{n-1}\eta(2k)\zeta(2n-2k+1), \ n\ge1. \end{equation*}

may be found in Cornel's new article here.

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  • $\begingroup$ "At a later date" is not really good. In particular, how to you go from $\sum _{n=1} ^{\infty} (-1)^{n-1} \frac {H_n} n$ to $\sum _{k=1} ^{\infty} (-1)^{k-1} \frac {H_k} {k^{2n}}$? $\endgroup$
    – Alex M.
    May 23, 2019 at 14:31
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For $A(1,2)$:

Using $$\sum_{n=1}^\infty H_nx^n=-\frac{\ln(1-x)}{1-x}$$

replace $x$ with $-x$ then multiply both sides by $-\frac{\ln x}{x}$ and integrate between $0$ and $1$ and use the fact that $\int_0^1 -x^{n-1}\ln xdx=\frac{1}{n^2}$ we get

$$\sum_{n=1}^\infty\frac{(-1)^{n}H_n}{n^2}=\int_0^1\frac{\ln x\ln(1+x)}{x(1+x)}dx$$

$$=\underbrace{\int_0^1\frac{\ln x\ln(1+x)}{x}dx}_{IBP}-\underbrace{\int_0^1\frac{\ln x\ln(1+x)}{1+x}dx}_{IBP}$$

$$=\int_0^1\frac{\operatorname{Li}_2(-x)}{x}dx+\frac12\int_0^1\frac{\ln^2(1+x)}{x}dx$$

$$=-\frac34\zeta(3)+\frac12\left(\frac14\zeta(3)\right)=\boxed{-\frac58\zeta(3)}$$


Proof of $\int_0^1\frac{\ln^2(1+x)}{x}dx$:

Proof 1:

Using the algebraic identity

$$b^2=\frac12(a-b)^2+\frac12(a+b)^2-a^2$$

let $a=\ln(1-x)$ and $b=\ln(1+x)$ we have

$$\int_0^1\frac{\ln^2(1+x)}{x}\ dx=\frac12\underbrace{\int_0^1\frac{\ln^2\left(\frac{1-x}{1+x}\right)}{x}\ dx}_{\frac{1-x}{1+x}=y}+\frac12\underbrace{\int_0^1\frac{\ln^2(1-x^2)}{x}\ dx}_{1-x^2=y}-\underbrace{\int_0^1\frac{\ln^2(1-x)}{x}\ dx}_{1-x=y}\\=\int_0^1\frac{\ln^2y}{1-y^2}\ dy+\frac14\int_0^1\frac{\ln^2y}{1-y}\ dy-\int_0^1\frac{\ln^2y}{1-y}\ dy\\=\frac12\int_0^1\frac{\ln^2y}{1+y}\ dy-\frac14\int_0^1\frac{\ln^2y}{1-y}\ dy=\frac12\left(\frac32\zeta(3)\right)-\frac14(2\zeta(3))=\boxed{\frac14\zeta(3)}$$

Proof 2:

Using the generalization

$$\int_0^1\frac{\ln^n(1+x)}{x}dx=\frac{\ln^{n+1}(2)}{n+1}+n!\zeta(n+1)+\sum_{k=0}^n k!{n\choose k}\ln^{n-k}(2)\operatorname{Li}_{k+1}\left(\frac12\right)$$


For $A(2,1)$:

By Cauchy product we have

$$-\ln(1-x)\operatorname{Li}_2(x)=\sum_{n=1}^\infty\left(\frac{2H_n}{n^2}+\frac{H_n^{(2)}}{n}-\frac3{n^3}\right)x^n$$

Set $x=-1$ and rearrange we get

$$\sum_{n=1}^\infty\frac{(-1)^nH_n^{(2)}}{n}=3\operatorname{Li}_3(-1)-\ln(2)\operatorname{Li}_2(-1)-2\sum_{n=1}^\infty\frac{(-1)^{n}H_n}{n^2}$$

$$=3\left(-\frac34\zeta(3)\right)-\ln(2)\left(-\frac12\zeta(2)\right)-2\left(-\frac58\zeta(3)\right)=\boxed{\frac12\ln(2)\zeta(2)-\zeta(3)}$$

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For convenience define, $$S(m,p)=\sum_{(a,b)\in \mathbb{N^2}}\frac{(-1)^{a+b}}{a^m(a+b)^p}$$

So that,

$$S(m,p)+A(m,p)=\eta(m+p)$$

Where $\eta$ is the dirichlet eta function

Now since, $$\sum_{j=1}^{k-1}\frac{1}{a^j(a+b)^{k-j}}=\frac{a}{ba^k}-\frac{a}{b(a+b)^k}-\frac{1}{(a+b)^k}$$

We get the reccurence relation,

$$\sum_{j=1}^{k-1}A(j,k-j)=k\eta(k)-\ln(2)\eta(k-1)-A(1,k-1)$$

From which we get the value of $A(1,1)$

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Let us start by noting that the first two sums below are the same (interchange the summation variables and the order of the sums) \begin{eqnarray*} \sum_{n=1}^{\infty} \sum_{m=1}^{\infty} \frac{(-1)^{n+m}}{n(n+m)} + \sum_{n=1}^{\infty} \sum_{m=1}^{\infty} \frac{(-1)^{n+m}}{m(n+m)} = \left( \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n} \right) \left( \sum_{m=1}^{\infty} \frac{(-1)^{m+1}}{m} \right). \end{eqnarray*} Thus, we have \begin{eqnarray*} \sum_{n=1}^{\infty} \sum_{m=1}^{\infty} \frac{(-1)^{n+m}}{n(n+m)} = \frac{(\ln(2))^2}{2}. \end{eqnarray*} Now \begin{eqnarray*} A(1,1) = \sum_{k=1}^{\infty} \frac{(-1)^{k+1}H_k}{k} &=& \sum_{k=1}^{\infty} \frac{(-1)^{k+1}}{k^2} - \sum_{n=1}^{\infty} \sum_{m=1}^{\infty} \frac{(-1)^{n+m}}{n(n+m)} \\ &=& \frac{1}{2} \zeta_2 - \frac{1}{2} ( \ln(2) )^2. \end{eqnarray*}

Consider the Harmonic numbers in two ways \begin{eqnarray*} H_n=\sum_{k=1}^{n} \frac{1}{k} = \sum_{m=1}^{\infty} \left( \frac{1}{m} -\frac{1}{m+n} \right). \end{eqnarray*} We have \begin{eqnarray*} \sum_{k=1}^{\infty} \frac{H_k}{k^2} &=& \sum_{m=1}^{\infty} \frac{1}{m^3} + \sum_{n=1}^{\infty} \sum_{m=1}^{\infty} \frac{1}{n(n+m)^2} \\ &=& \sum_{n=1}^{\infty} \sum_{m=1}^{\infty} \frac{1}{nm(n+m)} . \end{eqnarray*} As we saw earlier, the first two sums below are the same (interchange the summation variables and the order of the sums) \begin{eqnarray*} \sum_{n=1}^{\infty} \sum_{m=1}^{\infty} \frac{1}{n(n+m)^2} + \sum_{n=1}^{\infty} \sum_{m=1}^{\infty} \frac{1}{m(n+m)^2} = \sum_{n=1}^{\infty} \sum_{m=1}^{\infty} \frac{1}{nm(n+m)}. \end{eqnarray*} After a little bit of algebra \begin{eqnarray*} \sum_{n=1}^{\infty} \sum_{m=1}^{\infty} \frac{1}{nm(n+m)} = 2 \zeta_3 \\ \sum_{n=1}^{\infty} \sum_{m=1}^{\infty} \frac{1}{n(n+m)^2} = \zeta_3 . \\ \end{eqnarray*} Next, split the sum $\sum_{n=1}^{\infty} \sum_{m=1}^{\infty} \frac{1}{nm(n+m)}$ according to weather $m>n,m=n$ and $m<n$, this gives \begin{eqnarray*} \sum_{n=1}^{\infty} \sum_{m=1}^{\infty} \frac{1}{nm(n+m)} = 2 \sum_{n=1}^{\infty} \sum_{m=1}^{\infty} \frac{1}{n(n+m)(2n+m)} +\frac{1}{2}\sum_{n=1}^{\infty} \frac{1}{n^3} \end{eqnarray*} So \begin{eqnarray*} \sum_{n=1}^{\infty} \sum_{m=1}^{\infty} \frac{1}{n(n+m)(2n+m)} = \frac{3}{4} \zeta_3. \end{eqnarray*} Partial fractions ... \begin{eqnarray*} \underbrace{\sum_{n=1}^{\infty} \sum_{m=1}^{\infty} \frac{1}{nm(n+m)}}_{2 \zeta_3} + \underbrace{\sum_{n=1}^{\infty} \sum_{m=1}^{\infty} \frac{1}{n(n+m)(2n+m)}}_{\frac{3}{4} \zeta_3} = 2 \sum_{n=1}^{\infty} \sum_{m=1}^{\infty} \frac{1}{nm(2n+m)} \end{eqnarray*} gives \begin{eqnarray*} \sum_{n=1}^{\infty} \sum_{m=1}^{\infty} \frac{1}{nm(2n+m)} = \frac{11}{8} \zeta_3. \end{eqnarray*} Partial fractions ... \begin{eqnarray*} \underbrace{\sum_{n=1}^{\infty} \sum_{m=1}^{\infty} \frac{1}{nm(2n+m)}}_{\frac{11}{8} \zeta_3} + \sum_{n=1}^{\infty} \sum_{m=1}^{\infty} \frac{1}{m(n+m)(2n+m)} = \underbrace{\sum_{n=1}^{\infty} \sum_{m=1}^{\infty} \frac{1}{nm(n+m)}}_{2 \zeta_3} \end{eqnarray*} gives \begin{eqnarray*} \sum_{n=1}^{\infty} \sum_{m=1}^{\infty} \frac{1}{m(n+m)(2n+m)} = \frac{5}{8} \zeta_3. \end{eqnarray*} Next, consider the sum $ \sum_{n=1}^{\infty} \sum_{m=1}^{\infty} \frac{1}{n(n+m)(2n+m)} $ according to weather $m$ is odd or even \begin{eqnarray*} \underbrace{\sum_{n=1}^{\infty} \sum_{m=1}^{\infty} \frac{1}{n(n+m)(2n+m)}}_{ \frac{3}{4} \zeta_3} = \sum_{n=1}^{\infty} \sum_{m=1}^{\infty} \frac{1}{n(2m-1)(2n+2m-1)}+ \frac{1}{2} \underbrace{ \sum_{n=1}^{\infty} \sum_{m=1}^{\infty} \frac{1}{n(n+m)(n+2m)}}_{ \frac{5}{8} \zeta_3} \end{eqnarray*} so \begin{eqnarray*} \sum_{n=1}^{\infty} \sum_{m=1}^{\infty} \frac{1}{n(2m-1)(2n+2m-1)}= \frac{7}{16} \zeta_3. \end{eqnarray*} Again consider weather $m$ is odd or even \begin{eqnarray*} \sum_{n=1}^{\infty} \sum_{m=1}^{\infty} \frac{(-1)^{m+1}}{n(n+m)(2n+m)} = \underbrace{\sum_{n=1}^{\infty} \sum_{m=1}^{\infty} \frac{1}{n(2m-1)(2n+2m-1)}}_{ \frac{7}{16} \zeta_3} - \frac{1}{2} \underbrace{ \sum_{n=1}^{\infty} \sum_{m=1}^{\infty} \frac{1}{n(n+m)(n+2m)}}_{ \frac{5}{8} \zeta_3} \end{eqnarray*} so \begin{eqnarray*} \sum_{n=1}^{\infty} \sum_{m=1}^{\infty} \frac{(-1)^{m+1}}{n(n+m)(2n+m)}= \frac{1}{8} \zeta_3. \end{eqnarray*} Now, split the sum $\sum_{n=1}^{\infty} \sum_{m=1}^{\infty} \frac{(-1)^{m+n}}{nm(n+m)}$ according to weather $m>n,m=n$ and $m<n$, \begin{eqnarray*} \sum_{n=1}^{\infty} \sum_{m=1}^{\infty} \frac{(-1)^{m+n}}{nm(n+m)} = -2 \sum_{n=1}^{\infty} \sum_{m=1}^{\infty} \frac{(-1)^{m+1}}{n(n+m)(2n+m)} +\frac{1}{2}\sum_{n=1}^{\infty} \frac{1}{n^3} \end{eqnarray*} So \begin{eqnarray*} \sum_{n=1}^{\infty} \sum_{m=1}^{\infty} \frac{(-1)^{m+n}}{nm(n+m)} = \frac{1}{4} \zeta_3. \end{eqnarray*} Again, the first two sums below are equal \begin{eqnarray*} \sum_{n=1}^{\infty} \sum_{m=1}^{\infty} \frac{(-1)^{n+m}}{n(n+m)^2} + \sum_{n=1}^{\infty} \sum_{m=1}^{\infty} \frac{(-1)^{n+m}}{m(n+m)^2} = \sum_{n=1}^{\infty} \sum_{m=1}^{\infty} \frac{(-1)^{n+m}}{nm(n+m)} \end{eqnarray*} so \begin{eqnarray*} \sum_{n=1}^{\infty} \sum_{m=1}^{\infty} \frac{(-1)^{n+m}}{n(n+m)^2} = \frac{1}{8} \zeta_3. \end{eqnarray*} Note that \begin{eqnarray*} \sum_{n=1}^{\infty} \sum_{m=1}^{\infty} \frac{(-1)^{n+m}}{n^2(n+m)} + \underbrace{\sum_{n=1}^{\infty} \sum_{m=1}^{\infty} \frac{(-1)^{n+m}}{nm(n+m)}}_{\frac{1}{4} \zeta_3} = \underbrace{\left( \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n^2} \right) }_{\frac{1}{2} \zeta_2 } \underbrace{\left( \sum_{m=1}^{\infty} \frac{(-1)^{m+1}}{m} \right)}_{\ln(2)}. \end{eqnarray*} Thus, we have \begin{eqnarray*} \sum_{n=1}^{\infty} \sum_{m=1}^{\infty} \frac{(-1)^{n+m}}{n^2(n+m)} = \frac{1}{2} \zeta_2 \ln(2) - \frac{1}{4} \zeta_3. \end{eqnarray*}

So ... finally ... \begin{eqnarray*} A(1,2) = \sum_{k=1}^{\infty} \frac{(-1)^{k+1}H_k}{k^2} &=& \sum_{k=1}^{\infty} \frac{(-1)^{k+1}}{k^3} - \sum_{n=1}^{\infty} \sum_{m=1}^{\infty} \frac{(-1)^{n+m}}{n(n+m)^2} \\ &=& \frac{5}{8} \zeta_3 \end{eqnarray*} and \begin{eqnarray*} A(2,1) = \sum_{k=1}^{\infty} \frac{(-1)^{k+1}H_k^{(2)}}{k} &=& \sum_{k=1}^{\infty} \frac{(-1)^{k+1}}{k^3} - \sum_{n=1}^{\infty} \sum_{m=1}^{\infty} \frac{(-1)^{n+m}}{n^2(n+m)} \\ &=& \frac{1}{2} \zeta_3 - \frac{1}{2} \zeta_2 \ln(2). \end{eqnarray*}

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$$\begin{array}{l}{I_1} = \int\limits_0^1 {\frac{{x\ln \left( {1 - x} \right)}}{{1 + {x^2}}}\,dx\,\,\,,\,\,\,{I_2} = } \int\limits_0^1 {\frac{{x\ln \left( {1 + x} \right)}}{{1 + {x^2}}}\,dx\,\,\,} \\{I_1}\left( m \right) = \frac{1}{2}\int\limits_0^1 {\frac{{2x{{\left( {1 - x} \right)}^m}}}{{1 + {x^2}}}dx\,\,\,\,,\,{I_1}^{\rm{\backslash }}\left( {m = 0} \right) = I\,\,\,\,\,,{I_1}\left( m \right) = \left( {\frac{1}{2}{{\left( {1 - x} \right)}^m}\ln \left( {1 + {x^2}} \right)} \right)} _0^1 + \frac{m}{2}\int\limits_0^1 {\ln \left( {1 + {x^2}} \right)} {\left( {1 - x} \right)^{m - 1}}dx\\ = - \frac{m}{2}\sum\limits_{k = 1}^\infty {\frac{{{{\left( { - 1} \right)}^k}}}{k}\int\limits_0^1 {{x^{2k}}{{\left( {1 - x} \right)}^{m - 1}}dx = } } - \frac{m}{2}\sum\limits_{k = 1}^\infty {\frac{{{{\left( { - 1} \right)}^k}}}{k}\left( {\frac{{\Gamma \left( {2k + 1} \right)\Gamma \left( m \right)}}{{\Gamma \left( {2k + m + 1} \right)}}} \right)} \\\frac{{d{I_1}\left( m \right)}}{{dm}} = - \frac{1}{2}\sum\limits_{k = 1}^\infty {\frac{{{{\left( { - 1} \right)}^k}\Gamma \left( {2k + 1} \right)}}{k}\left( {\frac{{\Gamma \left( {2k + m + 1} \right){\Gamma ^{\rm{\backslash }}}\left( {m + 1} \right) - \Gamma \left( {m + 1} \right){\Gamma ^{\rm{\backslash }}}\left( {2k + m + 1} \right)}}{{{\Gamma ^2}\left( {2k + m + 1} \right)}}} \right)} \end{array} % MathType!MTEF!2!1!+- % feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaWGjb % WaaSbaaSqaaiaaigdaaeqaaOGaeyypa0Zaa8qCaeaadaWcaaqaaiaa % dIhaciGGSbGaaiOBamaabmaabaGaaGymaiabgkHiTiaadIhaaiaawI % cacaGLPaaaaeaacaaIXaGaey4kaSIaamiEamaaCaaaleqabaGaaGOm % aaaaaaGccaaMc8UaamizaiaadIhacaaMc8UaaGPaVlaaykW7caGGSa % GaaGPaVlaaykW7caaMc8UaamysamaaBaaaleaacaaIYaaabeaakiab % g2da9aWcbaGaaGimaaqaaiaaigdaa0Gaey4kIipakmaapehabaWaaS % aaaeaacaWG4bGaciiBaiaac6gadaqadaqaaiaaigdacqGHRaWkcaWG % 4baacaGLOaGaayzkaaaabaGaaGymaiabgUcaRiaadIhadaahaaWcbe % qaaiaaikdaaaaaaOGaaGPaVlaadsgacaWG4bGaaGPaVlaaykW7caaM % c8oaleaacaaIWaaabaGaaGymaaqdcqGHRiI8aaGcbaGaamysamaaBa % aaleaacaaIXaaabeaakmaabmaabaGaamyBaaGaayjkaiaawMcaaiab % g2da9maalaaabaGaaGymaaqaaiaaikdaaaWaa8qCaeaadaWcaaqaai % aaikdacaWG4bWaaeWaaeaacaaIXaGaeyOeI0IaamiEaaGaayjkaiaa % wMcaamaaCaaaleqabaGaamyBaaaaaOqaaiaaigdacqGHRaWkcaWG4b % WaaWbaaSqabeaacaaIYaaaaaaakiaadsgacaWG4bGaaGPaVlaaykW7 % caaMc8UaaGPaVlaacYcacaaMc8UaamysamaaBaaaleaacaaIXaaabe % aakmaaCaaaleqabaGaafixaaaakmaabmaabaGaamyBaiabg2da9iaa % icdaaiaawIcacaGLPaaacqGH9aqpcaWGjbGaaGPaVlaaykW7caaMc8 % UaaGPaVlaaykW7caGGSaGaamysamaaBaaaleaacaaIXaaabeaakmaa % bmaabaGaamyBaaGaayjkaiaawMcaaiabg2da9maabmaabaWaaSaaae % aacaaIXaaabaGaaGOmaaaadaqadaqaaiaaigdacqGHsislcaWG4baa % caGLOaGaayzkaaWaaWbaaSqabeaacaWGTbaaaOGaciiBaiaac6gada % qadaqaaiaaigdacqGHRaWkcaWG4bWaaWbaaSqabeaacaaIYaaaaaGc % caGLOaGaayzkaaaacaGLOaGaayzkaaaaleaacaaIWaaabaGaaGymaa % qdcqGHRiI8aOWaa0baaSqaaiaaicdaaeaacaaIXaaaaOGaey4kaSYa % aSaaaeaacaWGTbaabaGaaGOmaaaadaWdXbqaaiGacYgacaGGUbWaae % WaaeaacaaIXaGaey4kaSIaamiEamaaCaaaleqabaGaaGOmaaaaaOGa % ayjkaiaawMcaaaWcbaGaaGimaaqaaiaaigdaa0Gaey4kIipakmaabm % aabaGaaGymaiabgkHiTiaadIhaaiaawIcacaGLPaaadaahaaWcbeqa % aiaad2gacqGHsislcaaIXaaaaOGaamizaiaadIhaaeaacqGH9aqpcq % GHsisldaWcaaqaaiaad2gaaeaacaaIYaaaamaaqahabaWaaSaaaeaa % daqadaqaaiabgkHiTiaaigdaaiaawIcacaGLPaaadaahaaWcbeqaai % aadUgaaaaakeaacaWGRbaaamaapehabaGaamiEamaaCaaaleqabaGa % aGOmaiaadUgaaaGcdaqadaqaaiaaigdacqGHsislcaWG4baacaGLOa % GaayzkaaWaaWbaaSqabeaacaWGTbGaeyOeI0IaaGymaaaakiaadsga % caWG4bGaeyypa0daleaacaaIWaaabaGaaGymaaqdcqGHRiI8aaWcba % Gaam4Aaiabg2da9iaaigdaaeaacqGHEisPa0GaeyyeIuoakiabgkHi % TmaalaaabaGaamyBaaqaaiaaikdaaaWaaabCaeaadaWcaaqaamaabm % aabaGaeyOeI0IaaGymaaGaayjkaiaawMcaamaaCaaaleqabaGaam4A % aaaaaOqaaiaadUgaaaWaaeWaaeaadaWcaaqaaiabfo5ahnaabmaaba % GaaGOmaiaadUgacqGHRaWkcaaIXaaacaGLOaGaayzkaaGaeu4KdC0a % aeWaaeaacaWGTbaacaGLOaGaayzkaaaabaGaeu4KdC0aaeWaaeaaca % aIYaGaam4AaiabgUcaRiaad2gacqGHRaWkcaaIXaaacaGLOaGaayzk % aaaaaaGaayjkaiaawMcaaaWcbaGaam4Aaiabg2da9iaaigdaaeaacq % GHEisPa0GaeyyeIuoaaOqaamaalaaabaGaamizaiaadMeadaWgaaWc % baGaaGymaaqabaGcdaqadaqaaiaad2gaaiaawIcacaGLPaaaaeaaca % WGKbGaamyBaaaacqGH9aqpcqGHsisldaWcaaqaaiaaigdaaeaacaaI % YaaaamaaqahabaWaaSaaaeaadaqadaqaaiabgkHiTiaaigdaaiaawI % cacaGLPaaadaahaaWcbeqaaiaadUgaaaGccqqHtoWrdaqadaqaaiaa % ikdacaWGRbGaey4kaSIaaGymaaGaayjkaiaawMcaaaqaaiaadUgaaa % WaaeWaaeaadaWcaaqaaiabfo5ahnaabmaabaGaaGOmaiaadUgacqGH % RaWkcaWGTbGaey4kaSIaaGymaaGaayjkaiaawMcaaiabfo5ahnaaCa % aaleqabaGaagixaaaakmaabmaabaGaamyBaiabgUcaRiaaigdaaiaa % wIcacaGLPaaacqGHsislcqqHtoWrdaqadaqaaiaad2gacqGHRaWkca % aIXaaacaGLOaGaayzkaaGaeu4KdC0aaWbaaSqabeaacayGCbaaaOWa % aeWaaeaacaaIYaGaam4AaiabgUcaRiaad2gacqGHRaWkcaaIXaaaca % GLOaGaayzkaaaabaGaeu4KdC0aaWbaaSqabeaacaaIYaaaaOWaaeWa % aeaacaaIYaGaam4AaiabgUcaRiaad2gacqGHRaWkcaaIXaaacaGLOa % GaayzkaaaaaaGaayjkaiaawMcaaaWcbaGaam4Aaiabg2da9iaaigda % aeaacqGHEisPa0GaeyyeIuoaaaaa!54A6! $$ $$\begin{array}{l}{I_1} = \frac{1}{2}\sum\limits_{k = 1}^\infty {\frac{{{{\left( { - 1} \right)}^k}}}{k}} \left( {\gamma + \psi \left( {2k + 1} \right)} \right) = \frac{1}{2}\sum\limits_{k = 1}^\infty {\frac{{{{\left( { - 1} \right)}^k}}}{k}} {H_{2k}}\\from\,\,\int\limits_0^1 {{x^{k - 1}}\ln \left( {1 - x} \right)\,dx = - \frac{{{H_k}}}{k}} \\\sum\limits_{k = 1}^\infty {\frac{{{{\left( { - 1} \right)}^k}}}{{2k}}} {H_{2k}} - \sum\limits_{k = 1}^\infty {\frac{{{{\left( { - 1} \right)}^k}}}{{2k}}} {H_k} = \sum\limits_{k = 1}^\infty {{{\left( { - 1} \right)}^k}} \int\limits_0^1 {{x^{2k - 1}}\ln \left( {1 + x} \right)\,dx} = \int\limits_0^1 {\frac{{x\ln \left( {1 + x} \right)}}{{1 + {x^2}}}\,dx\, = {I_2}\,\,} \\{I_1} + {I_2} = \sum\limits_{k = 1}^\infty {\frac{{{{\left( { - 1} \right)}^k}}}{{2k}}} {H_k} \Rightarrow \left( 1 \right)\end{array} % MathType!MTEF!2!1!+- % feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaWGjb % WaaSbaaSqaaiaaigdaaeqaaOGaeyypa0ZaaSaaaeaacaaIXaaabaGa % aGOmaaaadaaeWbqaamaalaaabaWaaeWaaeaacqGHsislcaaIXaaaca % GLOaGaayzkaaWaaWbaaSqabeaacaWGRbaaaaGcbaGaam4AaaaaaSqa % aiaadUgacqGH9aqpcaaIXaaabaGaeyOhIukaniabggHiLdGcdaqada % qaaiabeo7aNjabgUcaRiabeI8a5naabmaabaGaaGOmaiaadUgacqGH % RaWkcaaIXaaacaGLOaGaayzkaaaacaGLOaGaayzkaaGaeyypa0ZaaS % aaaeaacaaIXaaabaGaaGOmaaaadaaeWbqaamaalaaabaWaaeWaaeaa % cqGHsislcaaIXaaacaGLOaGaayzkaaWaaWbaaSqabeaacaWGRbaaaa % GcbaGaam4AaaaaaSqaaiaadUgacqGH9aqpcaaIXaaabaGaeyOhIuka % niabggHiLdGccaWGibWaaSbaaSqaaiaaikdacaWGRbaabeaaaOqaai % aadAgacaWGYbGaam4Baiaad2gacaaMc8UaaGPaVpaapehabaGaamiE % amaaCaaaleqabaGaam4AaiabgkHiTiaaigdaaaGcciGGSbGaaiOBam % aabmaabaGaaGymaiabgkHiTiaadIhaaiaawIcacaGLPaaacaaMc8Ua % amizaiaadIhacqGH9aqpcqGHsisldaWcaaqaaiaadIeadaWgaaWcba % Gaam4AaaqabaaakeaacaWGRbaaaaWcbaGaaGimaaqaaiaaigdaa0Ga % ey4kIipaaOqaamaaqahabaWaaSaaaeaadaqadaqaaiabgkHiTiaaig % daaiaawIcacaGLPaaadaahaaWcbeqaaiaadUgaaaaakeaacaaIYaGa % am4AaaaaaSqaaiaadUgacqGH9aqpcaaIXaaabaGaeyOhIukaniabgg % HiLdGccaWGibWaaSbaaSqaaiaaikdacaWGRbaabeaakiabgkHiTmaa % qahabaWaaSaaaeaadaqadaqaaiabgkHiTiaaigdaaiaawIcacaGLPa % aadaahaaWcbeqaaiaadUgaaaaakeaacaaIYaGaam4AaaaaaSqaaiaa % dUgacqGH9aqpcaaIXaaabaGaeyOhIukaniabggHiLdGccaWGibWaaS % baaSqaaiaadUgaaeqaaOGaeyypa0ZaaabCaeaadaqadaqaaiabgkHi % TiaaigdaaiaawIcacaGLPaaadaahaaWcbeqaaiaadUgaaaaabaGaam % 4Aaiabg2da9iaaigdaaeaacqGHEisPa0GaeyyeIuoakmaapehabaGa % aiiEamaaCaaaleqabaGaaGOmaiaadUgacqGHsislcaaIXaaaaOGaci % iBaiaac6gadaqadaqaaiaaigdacqGHRaWkcaWG4baacaGLOaGaayzk % aaGaaGPaVlaadsgacaWG4baaleaacaaIWaaabaGaaGymaaqdcqGHRi % I8aOGaeyypa0Zaa8qCaeaadaWcaaqaaiaadIhaciGGSbGaaiOBamaa % bmaabaGaaGymaiabgUcaRiaadIhaaiaawIcacaGLPaaaaeaacaaIXa % Gaey4kaSIaamiEamaaCaaaleqabaGaaGOmaaaaaaGccaaMc8Uaamiz % aiaadIhacaaMc8Uaeyypa0JaamysamaaBaaaleaacaaIYaaabeaaki % aaykW7caaMc8oaleaacaaIWaaabaGaaGymaaqdcqGHRiI8aaGcbaGa % amysamaaBaaaleaacaaIXaaabeaakiabgUcaRiaadMeadaWgaaWcba % GaaGOmaaqabaGccqGH9aqpdaaeWbqaamaalaaabaWaaeWaaeaacqGH % sislcaaIXaaacaGLOaGaayzkaaWaaWbaaSqabeaacaWGRbaaaaGcba % GaaGOmaiaadUgaaaaaleaacaWGRbGaeyypa0JaaGymaaqaaiabg6Hi % LcqdcqGHris5aOGaamisamaaBaaaleaacaWGRbaabeaakiabgkDiEp % aabmaabaGaaGymaaGaayjkaiaawMcaaaaaaa!EDD5! $$ $$\begin{array}{l}{I_1} + {I_2} = \int\limits_0^1 {\frac{{x\ln \left( {1 - {x^2}} \right)}}{{1 + {x^2}}}\,dx\,\,\,} = \frac{1}{2}\int\limits_0^1 {\frac{{\ln \left( {1 - x} \right)}}{{1 + x}}\,dx\,\,\,} \\by\,\,\,using\,\,the\,\,x = \frac{{1 - y}}{{1 + y}}\,\,\,\,\\\sum\limits_{k = 1}^\infty {\frac{{{{\left( { - 1} \right)}^k}}}{{2k}}} {H_k} = {I_1} + {I_2} = \frac{1}{2}\int\limits_0^1 {\frac{{\ln \left( {\frac{{2y}}{{1 + y}}} \right)}}{{1 + y}}\,dy = \frac{1}{4}{{\ln }^2}2 - \frac{{{\pi ^2}}}{{24}}} \\and\,\,then\,\,we\,\,obtain\,\,the\,\,needed\,\,proof\end{array} % MathType!MTEF!2!1!+- % feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaWGjb % WaaSbaaSqaaiaaigdaaeqaaOGaey4kaSIaamysamaaBaaaleaacaaI % Yaaabeaakiabg2da9maapehabaWaaSaaaeaacaWG4bGaciiBaiaac6 % gadaqadaqaaiaaigdacqGHsislcaWG4bWaaWbaaSqabeaacaaIYaaa % aaGccaGLOaGaayzkaaaabaGaaGymaiabgUcaRiaadIhadaahaaWcbe % qaaiaaikdaaaaaaOGaaGPaVlaadsgacaWG4bGaaGPaVlaaykW7caaM % c8oaleaacaaIWaaabaGaaGymaaqdcqGHRiI8aOGaeyypa0ZaaSaaae % aacaaIXaaabaGaaGOmaaaadaWdXbqaamaalaaabaGaciiBaiaac6ga % daqadaqaaiaaigdacqGHsislcaWG4baacaGLOaGaayzkaaaabaGaaG % ymaiabgUcaRiaadIhaaaGaaGPaVlaadsgacaWG4bGaaGPaVlaaykW7 % caaMc8oaleaacaaIWaaabaGaaGymaaqdcqGHRiI8aaGcbaGaamOyai % aadMhacaaMc8UaaGPaVlaaykW7caWG1bGaam4CaiaadMgacaWGUbGa % am4zaiaaykW7caaMc8UaamiDaiaadIgacaWGLbGaaGPaVlaaykW7ca % WG4bGaeyypa0ZaaSaaaeaacaaIXaGaeyOeI0IaamyEaaqaaiaaigda % cqGHRaWkcaWG5baaaiaaykW7caaMc8UaaGPaVlaaykW7aeaadaaeWb % qaamaalaaabaWaaeWaaeaacqGHsislcaaIXaaacaGLOaGaayzkaaWa % aWbaaSqabeaacaWGRbaaaaGcbaGaaGOmaiaadUgaaaaaleaacaWGRb % Gaeyypa0JaaGymaaqaaiabg6HiLcqdcqGHris5aOGaamisamaaBaaa % leaacaWGRbaabeaakiabg2da9iaadMeadaWgaaWcbaGaaGymaaqaba % GccqGHRaWkcaWGjbWaaSbaaSqaaiaaikdaaeqaaOGaeyypa0ZaaSaa % aeaacaaIXaaabaGaaGOmaaaadaWdXbqaamaalaaabaGaciiBaiaac6 % gadaqadaqaamaalaaabaGaaGOmaiaadMhaaeaacaaIXaGaey4kaSIa % amyEaaaaaiaawIcacaGLPaaaaeaacaaIXaGaey4kaSIaamyEaaaaca % aMc8UaamizaiaadMhacqGH9aqpdaWcaaqaaiaaigdaaeaacaaI0aaa % aiGacYgacaGGUbWaaWbaaSqabeaacaaIYaaaaOGaaGOmaiabgkHiTm % aalaaabaGaeqiWda3aaWbaaSqabeaacaaIYaaaaaGcbaGaaGOmaiaa % isdaaaaaleaacaaIWaaabaGaaGymaaqdcqGHRiI8aaGcbaGaamyyai % aad6gacaWGKbGaaGPaVlaaykW7caWG0bGaamiAaiaadwgacaWGUbGa % aGPaVlaaykW7caWG3bGaamyzaiaaykW7caaMc8Uaam4Baiaadkgaca % WG0bGaamyyaiaadMgacaWGUbGaaGPaVlaaykW7caWG0bGaamiAaiaa % dwgacaaMc8UaaGPaVlaad6gacaWGLbGaamyzaiaadsgacaWGLbGaam % izaiaaykW7caaMc8UaamiCaiaadkhacaWGVbGaam4BaiaadAgaaaaa % !ED28! $$ by Mr. Sherif Hamed

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This is a special case of Kouba's https://arxiv.org/abs/1010.1842 equation $$\sum_{n\ge 1}(-)^{n-1}\frac{H_{kn}}{n} = \frac{(k^2+1)\pi^2}{24k}-\frac12 \sum_{j=0}^{k-1}\log^2\left(2\sin\frac{(2j+1)\pi}{2k}\right) $$

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