# Proof of $\frac{1}{e^{\pi}+1}+\frac{3}{e^{3\pi}+1}+\frac{5}{e^{5\pi}+1}+\ldots=\frac{1}{24}$

I would like to prove that $\displaystyle\sum_{\substack{n=1\\n\text{ odd}}}^{\infty}\frac{n}{e^{n\pi}+1}=\frac1{24}$.

I found a solution by myself 10 hours after I posted it, here it is:

$$f(x)=\sum_{\substack{n=1\\n\text{ odd}}}^{\infty}\frac{nx^n}{1+x^n},\quad\quad g(x)=\displaystyle\sum_{n=1}^{\infty}\frac{nx^n}{1-x^n},$$

then I must prove that $f(e^{-\pi})=\frac1{24}$. It was not hard to find the relation between $f(x)$ and $g(x)$, namely $f(x)=g(x)-4g(x^2)+4g(x^4)$.

Note that $g(x)$ is a Lambert series, so by expanding the Taylor series for the denominators and reversing the two sums, I get

$$g(x)=\sum_{n=1}^{\infty}\sigma(n)x^n$$

where $\sigma$ is the divisor function $\sigma(n)=\sum_{d\mid n}d$.

I then define for complex $\tau$ the function $$G_2(\tau)=\frac{\pi^2}3\Bigl(1-24\sum_{n=1}^{\infty}\sigma(n)e^{2\pi in\tau}\Bigr)$$ so that $$f(e^{-\pi})=g(e^{-\pi})-4g(e^{-2\pi})+4g(e^{-4\pi})=\frac1{24}+\frac{-G_2(\frac i2)+4G_2(i)-4G_2(2i)}{8\pi^2}.$$

But it is proven in Apostol "Modular forms and Dirichlet Series", page 69-71 that $G_2\bigl(-\frac1{\tau}\bigr)=\tau^2G_2(\tau)-2\pi i\tau$, which gives $\begin{cases}G_2(i)=-G_2(i)+2\pi\\ G_2(\frac i2)=-4G_2(2i)+4\pi\end{cases}\quad$. This is exactly was needed to get the desired result.

Hitoshigoto oshimai !

I find that sum fascinating. $e,\pi$ all together to finally get a rational. This is why mathematics is beautiful!

Thanks to everyone who contributed.

• See the article on the odd divisor function on MathWorld. According to this article, your function $f(x)$ is $1/24$ multiplied by the sum of the fourth powers of two Jacobi theta functions. May 12, 2013 at 7:09
• @danodare: If you found an answer by yourself, you are welcome to answer your own question. May 12, 2013 at 22:30
• Whence did this question come?
– robjohn
Aug 21, 2013 at 21:34
• That is a great solution! Wow. Aug 31, 2014 at 17:39
• are you sure ?than the sum is equal to $$\frac{1}{24}$$ when x=1 look
– user167276
Dec 29, 2015 at 19:44

We will use the Mellin transform technique. Recalling the Mellin transform and its inverse

$$F(s) =\int_0^{\infty} x^{s-1} f(x)dx, \quad\quad f(x)=\frac{1}{2 \pi i} \int_{c-i \infty}^{c+i \infty} x^{-s} F(s)\, ds.$$

Now, let's consider the function

$$f(x)= \frac{x}{e^{\pi x}+1}.$$

Taking the Mellin transform of $f(x)$, we get

$$F(s)={\pi }^{-s-1}\Gamma \left( s+1 \right) \left(1- {2}^{-s} \right) \zeta \left( s+1 \right),$$

where $\zeta(s)$ is the zeta function . Representing the function in terms of the inverse Mellin Transform, we have

$$\frac{x}{e^{\pi x}+1}=\frac{1}{2\pi i}\int_{C}{\pi }^{-s-1}\Gamma \left( s+1 \right) \left( 1-{2}^{-s} \right) \zeta \left( s+1 \right) x^{-s}ds.$$

Substituting $x=2n+1$ and summing yields

$$\sum_{n=0}^{\infty}\frac{2n+1}{e^{\pi (2n+1)}+1}=\frac{1}{2\pi i}\int_{C}{\pi}^{-s-1}\Gamma \left( s+1 \right)\left(1-{2}^{-s} \right) \zeta\left( s+1 \right) \sum_{n=0}^{\infty}(2n+1)^{-s}ds$$

$$= \frac{1}{2\pi i}\int_{C}{\pi }^{-s-1}\Gamma \left( s+1 \right) \left(1-{2}^{-s} \right)^2\zeta\left( s+1 \right) \zeta(s)ds.$$

Now, the only contribution of the poles comes from the simple pole $s=1$ of $\zeta(s)$ and the residue equals to $\frac{1}{24}$. So, the sum is given by

$$\sum_{n=0}^{\infty}\frac{2n+1}{e^{\pi (2n+1)}+1}=\frac{1}{24}$$

Notes: 1)

$$\sum_{n=0}^{\infty}(2n+1)^{-s}= \left(1- {2}^{-s} \right) \zeta \left( s \right).$$

2) The residue of the simple pole $s=1$, which is the pole of the zeta function, can be calculated as

$$r = \lim_{s=1}(s-1)({\pi }^{-s-1}\Gamma \left( s+1 \right) \left({2}^{-s}-1 \right)^2\zeta\left( s+1 \right) \zeta(s))$$

$$= \lim_{s\to 1}(s-1)\zeta(s)\lim_{s\to 1} {\pi }^{-s-1}\Gamma \left( s+1 \right) \left({2}^{-s}-1 \right)^2\zeta\left( s+1 \right) = \frac{1}{24}.$$

For calculating the above limit, we used the facts

$$\lim_{s\to 1}(s-1)\zeta(s)=1, \quad \zeta(2)=\frac{\pi^2}{6}.$$

3) Here is the technique for computing the Mellin transform of $f(x)$.

• I have seen you use similar techniques many times before. Where did you learn them? You are quite good with integrals. May 12, 2013 at 7:32
• @Potato: These techniques for summing infinite series are known in the literature as applications of integral transforms such as Mellin and Fourier transform. I have been learning them over the years. They are very effective techniques. See here for a Fourier transform technique for summing a series. May 12, 2013 at 7:48
• Is it just me or is there an error in the signs of $$\sum_{k\ge 0} \frac{1}{(2k+1)^s} = \zeta(s) (1 - 2^{-s})$$ This error is repeated twice and canceled by the square, so that you still get the right answer. May 13, 2013 at 0:06
• This article is very nice work and if it should turn out that it does contain an error it definitely should be fixed so it can serve as a future reference to the method. May 13, 2013 at 1:13
• @MhenniBenghorbal I added a missing piece to your proof. Enjoy! We now have a complete example computation that we may link to in the future. May 13, 2013 at 23:36

Let's start with $$\sum_{n=0}^\infty x^n=\frac1{1-x}\tag{1}$$ Differentiating $(1)$ and multiplying by $x$, we get $$\sum_{n=0}^\infty nx^n=\frac{x}{(1-x)^2}\tag{2}$$ Taking the odd part of $(2)$ yields $$\sum_{n=0}^\infty(2n+1)x^{2n+1}=\frac{x(1+x^2)}{(1-x^2)^2}\tag{3}$$ Using $(3)$, we get \begin{align} \sum_{n=0}^\infty\frac{2n+1}{e^{(2n+1)\pi}+1} &=\sum_{n=0}^\infty\sum_{k=1}^\infty(-1)^{k-1}(2n+1)e^{-(2n+1)k\pi}\\ &=\sum_{k=1}^\infty\sum_{n=0}^\infty(-1)^{k-1}(2n+1)e^{-(2n+1)k\pi}\\ &=\sum_{k=1}^\infty(-1)^{k-1}\frac{e^{-k\pi}\left(1+e^{-2k\pi}\right)}{\left(1-e^{-2k\pi}\right)^2}\\ &=\frac12\sum_{k=1}^\infty(-1)^{k-1}\frac{\cosh(k\pi)}{\sinh^2(k\pi)}\tag{4} \end{align}

We can use the formula proven in this answer $$\pi\cot(\pi z)=\sum_{k\in\mathbb{Z}}\frac1{z+k}\tag{5}$$ to derive \begin{align} \pi\csc(\pi z) &=\pi\cot(\pi z/2)-\pi\cot(\pi z)\\[9pt] &=\sum_{k\in\mathbb{Z}}\frac2{z+2k}-\sum_{k\in\mathbb{Z}}\frac1{z+k}\\ &=\sum_{k\in\mathbb{Z}}\frac{(-1)^k}{z+k}\\ \pi^2\frac{\cos(\pi z)}{\sin^2(\pi z)} &=\sum_{k\in\mathbb{Z}}\frac{(-1)^k}{(z+k)^2}\tag{6} \end{align} then rotate coordinates with $z\mapsto iz$ to get $$\pi^2\frac{\cosh(\pi z)}{\sinh^2(\pi z)}=\sum_{j\in\mathbb{Z}}\frac{(-1)^j}{(z+ij)^2}\tag{7}$$

Now plug $(7)$ into $(4)$: \begin{align} \sum_{n=0}^\infty\frac{2n+1}{e^{(2n+1)\pi}+1} &=\frac1{2\pi^2}\sum_{k=1}^\infty\sum_{j\in\mathbb{Z}}(-1)^{j+k-1}\frac1{(k+ij)^2} \\ &=\frac1{2\pi^2}\sum_{k=1}^\infty(-1)^{k-1}\frac1{k^2}\\ &+\frac1{2\pi^2}\sum_{k=1}^\infty\sum_{j=1}^\infty(-1)^{j+k-1}\left(\frac1{(k+ij)^2}+\frac1{(k-ij)^2}\right)\\ &=\frac1{2\pi^2}\frac{\pi^2}{12}\\ &+\frac1{2\pi^2}\sum_{k=1}^\infty\sum_{j=1}^\infty(-1)^{j+k-1}\frac{2(k^2-j^2)}{(k^2+j^2)^2}\\ &=\frac1{24}+0\tag{8} \end{align}

• My bad; I didn't use the Mellin Transform. I just assumed $(1)$ and $(5)$ :-)
– robjohn
Aug 21, 2013 at 21:30
• ...or we could just notice that $\dfrac{\cosh}{\sinh^2}=-\bigg[\dfrac1{\sinh}\bigg]'$ Jan 23, 2015 at 20:37

The calculation of the Mellin transform of $f(x)$ is not present in the above answer, so I will show it here.

$$\mathfrak{M}\left(\frac{1}{e^{\pi x}+1};s \right) = \int_0^\infty \frac{1}{e^{\pi x}+1} x^{s-1} dx = \int_0^\infty \frac{1}{e^{\pi x}} \frac{1}{1+e^{-\pi x}} x^{s-1} dx \\= \int_0^\infty \frac{1}{e^{\pi x}} \sum_{q\ge 0} (-1)^q e^{-\pi q x} x^{s-1} dx = \int_0^\infty \sum_{q\ge 0} (-1)^q e^{-\pi (q+1) x} x^{s-1} dx \\ = \Gamma(s) \sum_{q\ge 0} (-1)^q \frac{1}{\pi^s (q+1)^s} = \frac{1}{\pi^s} \Gamma(s) (\zeta(s) - 2 \times 2^{-s} \times \zeta(s)) = \frac{1}{\pi^s} \Gamma(s) (1 - 2\times 2^{-s}) \zeta(s).$$ It now follows from the definition of the Mellin transform that $$\mathfrak{M}\left(\frac{x}{e^{\pi x}+1};s \right) = \frac{1}{\pi^{s+1}} \Gamma(s+1) (1 - 2^{-s}) \zeta(s+1).$$

• There is more material here and here and here. May 12, 2013 at 23:31
• One more thing -- another interesting example can be found here. May 12, 2013 at 23:39
• I apologize -- changed it. May 12, 2013 at 23:53
• doesn't "above answer" not always the above for everyone? Dec 28, 2014 at 5:52
• are you sure that the result 1/24 is correct look the postlink
– user167276
Dec 29, 2015 at 20:24

Actually the above is not quite complete, the missing piece is the proof that we can drop the contribution from the pole at $s=-1,$ which is $x/24.$ To verify this we have to show that $$\int_{-i\infty}^{i\infty} \frac{1}{\pi^{s+1}} \Gamma(s+1) (1-2^{-s})^2 \zeta(s+1)\zeta(s) ds = 0.$$ Now from the functional equation of the Riemann Zeta function we see that this integral is equal to $$-\int_{-i\infty}^{i\infty} \frac{\zeta(-s)}{\sin(1/2s\pi)} (2^s-1) (1-2^{-s}) \zeta(s) ds$$ Actually doing the accounting we find that the kernel $$g(s) = \frac{\zeta(-s)}{\sin(1/2s\pi)} (2^s-1) (1-2^{-s}) \zeta(s)$$ of this integral has the property that $g(s) = - g(-s)$ on the imaginary axis, so the integral is zero.

To see this consider what effect negation has on the individual terms. $$\zeta(-s)\zeta(s) \to \zeta(s)\zeta(-s),$$ $$(2^s-1)(1-2^{-s}) \to (2^{-s}-1)(1-2^s) = (2^s-1)(1-2^{-s}),$$ $$\sin(1/2 s\pi) \to \sin(1/2 (-s)\pi) = -\sin(1/2 s\pi).$$ The first two terms are even and the last one is odd, QED.

Note that we have taken advantage of the fact that $x=1$ ... for other values of $x$ this trick will not go through. Also relevant is that negation (rotation by 180 degrees about the origin) takes the imaginary axis to itself (this is not the case when we are integrating along some other line parallel to the imaginary axis in the right half plane).

• So the closed contour is a tall rectangle? May 15, 2013 at 20:16
• Yes, exactly, the two vertical sides are the lines $\Re(s)=3/2$ and $\Re(s) = 0.$ May 15, 2013 at 22:17
• Let me rephrase my questions. 1) Can the right side of the rectangle be any vertical line to the right of the line $\Re(s) =1$? 2) There is a simple pole at the origin (albeit with residue 0). Does the contour technically need to be indented? 3) Does the integral go to zero along the top and bottom of the rectangle since $|\Gamma(s)|$ decays quickly as $\Im(s)$ increases? May 16, 2013 at 0:10
• 1) Owing to the convergence of $\sum_{k\ge 0} \frac{1}{(2k+1)^s}$ in the half plane $\Re(s)>1$ and the fact that there are no additional poles the Mellin inversion integral can indeed be along any vertical line in that half plane. 2) The simple pole is cancelled by the $(1-2^{-s})$ term, no indentation necessary. 3) This is correct, the decrease is exponential. Finally, let me refer you to one of the experts on this one -- the paper "Mellin Transform and Its Applications" by Szpankowski on academia.edu contains many examples and is highly readable. (For some reason SE won't let me add a link.) May 16, 2013 at 1:21
• Thanks. If you have time, I started a related thread earlier today. math.stackexchange.com/questions/392706/… May 16, 2013 at 1:37

We can link the functions $g(x),g(x^{2}),g(x^{4})$ with elliptic integrals $K,E$ and modulus $k$. Also we switch to variable $q$ instead of $x$. The value of $q$ here is $q = e^{-\pi}$ so that $k = 1/\sqrt{2}$.

From this post we can see that \begin{align} g(q) &= \frac{1 - P(q)}{24} = \frac{1}{24}\left\{1 - \left(\frac{2K}{\pi}\right)^{2}\left(\frac{6E}{K} + k^{2} - 5\right)\right\}\tag{1}\\ g(q^{2}) &= \frac{1 - P(q^{2})}{24} = \frac{1}{24}\left\{1 - \left(\frac{2K}{\pi}\right)^{2}\left(\frac{3E}{K} + k^{2} - 2\right)\right\}\tag{2}\\ g(q^{4}) &= \frac{1 - P(q^{4})}{24} = \frac{1}{24}\left\{1 - \left(\frac{2K}{\pi}\right)^{2}\left(\frac{3E}{2K} + \frac{k^{2} - 2}{4}\right)\right\}\tag{3} \end{align} Using the above equations and the fact that $k = 1/\sqrt{2}$ we get $$f(q) = g(q) - 4g(q^{2}) + 4g(q^{4}) = \frac{1}{24}$$ Note: The formula for $P(q^{4})$ is not given in the linked post, but it can be obtained via the same technique as mentioned in the post, namely finding an expression for $\eta(q^{4})$ in terms of $K, k$ and then applying logarithmic differentiation.

Based on suggestion from robjohn (via comment) I add some details on the theory of elliptic integrals and theta functions so that the identities $(1), (2), (3)$ get some context.

Let us then assume that $k$ is a real number (which is called modulus) with $0 < k < 1$ and let $k' = \sqrt{1 - k^{2}}$ then we define elliptic integrals $$K(k) = \int_{0}^{\pi/2}\frac{dx}{\sqrt{1 - k^{2}\sin^{2}x}}, E(k) = \int_{0}^{\pi/2}\sqrt{1 - k^{2}\sin^{2}x}\,dx\tag{4}$$ The integrals $K(k), E(k), K(k'), E(k')$ are usually denoted by $K, E, K', E'$ when the argument $k, k'$ is available from context. These integrals satisfy the Legendre's Identity $$KE' + K'E - KK' = \frac{\pi}{2}\tag{5}$$ We have the derivatives $$\frac{dK}{dk} = \frac{E - k'^{2}K}{kk'^{2}},\,\,\frac{dE}{dk} = \frac{E - K}{k}\tag{6}$$ It is almost a piece of magic (created by mathemagician Jacobi) that it is possible to obtain the values of $k, k'$ from the values of $K, K'$ using another variable called nome defined by $$q = e^{-\pi K'/K}\tag{7}$$ The desired relations are given by theta functions $\vartheta_{2}(q), \vartheta_{3}(q), \vartheta_{4}(q)$ as follows $$k = \frac{\vartheta_{2}^{2}(q)}{\vartheta_{3}^{2}(q)},\, k' = \frac{\vartheta_{4}^{2}(q)}{\vartheta_{3}^{2}(q)},\, \frac{2K}{\pi} = \vartheta_{3}^{2}(q)\tag{8}$$ where \begin{align} \vartheta_{2}(q) &= \sum_{n = -\infty}^{\infty}q^{(n + 1/2)^{2}} = 2(q^{1/4} + q^{9/4} + q^{25/4} + \cdots )\notag\\ &= 2q^{1/4}\prod_{n = 1}^{\infty}(1 - q^{2n})(1 + q^{2n})^{2}\tag{9a}\\ \vartheta_{3}(q) &= \sum_{n = -\infty}^{\infty}q^{n^{2}} = 1 + 2q + 2q^{4} + 2q^{9} + \cdots\notag\\ &= \prod_{n = 1}^{\infty}(1 - q^{2n})(1 + q^{2n - 1})^{2}\tag{9b}\\ \vartheta_{4}(q) &= \sum_{n = -\infty}^{\infty}(-1)^{n}q^{n^{2}} = 1 - 2q + 2q^{4} - 2q^{9} + \cdots\notag\\ &= \prod_{n = 1}^{\infty}(1 - q^{2n})(1 - q^{2n - 1})^{2}\tag{9c} \end{align} Using the relations $(5), (6), (7)$ it is possible to show that $$\frac{dq}{dk} = \frac{\pi^{2}q}{2kk'^{2}K^{2}}\tag{10}$$ We next introduce the function $\eta(q)$ via $$\eta(q) = q^{1/24}\prod_{n = 1}^{\infty}(1 - q^{n})\tag{11}$$ Logarithmic differentiation of the above relation gives rise to the function $P(q)$ i.e. $$P(q) = 1 - 24\sum_{n = 1}^{\infty}\frac{nq^{n}}{1 - q^{n}} = 24q\frac{d}{dq}\log\eta(q)\tag{12}$$ Using equations $(8), (9a), (9b), (9c)$ (especially the product representations of theta functions) it is possible to express $\eta(q)$ in terms of $k, k', K$ and we have \begin{align} \eta(q) &= 2^{-1/6}\sqrt{\frac{2K}{\pi}}k^{1/12}k'^{1/3}\tag{13a}\\ \eta(q^{2}) &= 2^{-1/3}\sqrt{\frac{2K}{\pi}}(kk')^{1/6}\tag{13b}\\ \eta(q^{4}) &= 2^{-2/3}\sqrt{\frac{2K}{\pi}}k^{1/3}k'^{1/12}\tag{13c} \end{align} I obtained the expression of $\eta(q^{2})$ directly using the product representation of theta functions as $$2\{\eta(q^{2})\}^{3} = \vartheta_{2}(q)\vartheta_{3}(q)\vartheta_{4}(q)$$ and then equation $(13b)$ follows by using equation $(8)$. The expressions for $\eta(q), \eta(q^{4})$ were obtained by applying Landen's Transformation on these expressions (thus if we replace $k$ by $2\sqrt{k}/(1 + k)$ and $K$ by $(1 + k)K$ the variable $q^{2}$ is replaced by $q$ and if we replace $k$ by $(1 - k')/(1 + k')$ and $K$ by $(1 + k')K/2$ the variable $q$ is replaced by $q^{2}$). The identities $(1), (2)$ and $(3)$ of my answer are obtained by logarithmic differentiation of the above expressions for $\eta(q), \eta(q^{2}), \eta(q^{4})$.

• (+1) However, it would be nice to bring the information from the linked post here.
– robjohn
Mar 13, 2016 at 11:05
• @robjohn: I will post some details here but won't be possible to give entire information with full proof. Mar 13, 2016 at 14:11
• @robjohn: I have added some details regarding the identities linking elliptic integrals with theta functions. The eta function is also introduced in order to get the function $P(q)$. I have given links to my blog posts for more details and proofs. I hope this is useful. Mar 14, 2016 at 5:15
• Epic answer, which deserves a lot more upvotes (+1k) May 16, 2017 at 19:49
• @tired: thanks for encouraging words. Since I am not so much familiar with the powerful techniques of complex analysis, I try to deal with such identities via Ramanujan's techniques. May 16, 2017 at 22:57

$$S=\sum_{n=1}^{\infty }\frac{2n-1}{1+e^{\pi (2n-1)}},$$ first we know that $$\frac{1}{1+e^x}=\sum_{k=1}^{\infty }(-1)^{k-1}e^{-kx}$$ therefore $$S=\sum_{n=1}^{\infty }(2n-1)\sum_{k=1}^{\infty }(-1)^{k-1}e^{-k(2n-1)\pi }=\sum_{k=1}^{\infty }(-1)^{k-1}\sum_{n=1}^{\infty }(2n-1)e^{-k(2n-1)\pi }$$

but we know $$\sum_{n=1}^{\infty }e^{-(2n-1)\pi x}=\frac{1}{2sinh(\pi x)},$$ for all $$x>0$$ but $$\frac{1}{2sinh(\pi x)}=(\frac{i}{2\pi })(\frac{\pi }{sin(i\pi x)})$$ and $$\int_{0}^{\infty }\frac{t^{-x}}{1+t}dt=\frac{\pi }{sin(\pi x)}$$ therefore $$\int_{0}^{\infty }\frac{t^{-x}}{1+t}dt=\int_{0}^{1}\frac{t^{-x}}{1+t}dt+\int_{0}^{1}\frac{t^{x-1}}{1+t}dt=\int_{0}^{1}\frac{t^{-x}+t^{x-1}}{1+t}dt\\ \\ \frac{\pi }{sin(\pi x)}=\int_{0}^{1}\frac{t^{-x}+t^{x-1}}{1+t}dt=\sum_{k=1}^{\infty }(-1)^{k-1}(\frac{1}{k-x}+\frac{1}{k-1+x})\\ \therefore from\ \frac{\pi }{sin(i\pi x)}=\int_{0}^{\infty }\frac{t^{-ix}+t^{ix-1}}{1+t}dt=\sum_{k=1}^{\infty }(-1)^{k-1}(\frac{1}{k-ix}+\frac{1}{k-1+ix})\\ \\ =\sum_{k=1}^{\infty }(-1)^{k-1}(\frac{k+ix}{x^2+k^2}+\frac{k-1-ix}{(k-1^2)+x^2})=\sum_{k=1}^{\infty }(-1)^{k-1}(\frac{k}{k^2+x^2}+\frac{k-1}{x^2+(k-1)^2})\\ \\ +i\sum_{k=1}^{\infty }(-1)^{k-1}(\frac{x}{x^2+k^2}-\frac{x}{x^2+(k-1)^2})=i(\sum_{k=1}^{\infty }(-1)^{k-1}\frac{x}{x^2+k^2}-\sum_{k=1}^{\infty }(-1)^{k}\frac{x}{x^2+k^2})\\ \\ \\ =i[\frac{-1}{x}+\sum_{k=1}^{\infty }(-1)^{k-1}\frac{x}{x^2+k^2}+\sum_{k=1}^{\infty }(-1)^{k-1}\frac{x}{x^2+k^2}]=i[\frac{-1}{x}+2\sum_{k=1}^{\infty }(-1)^{k-1}\frac{x}{x^2+k^2}]\\ \\\therefore \frac{\pi }{sinh(\pi x)}=\frac{1}{x}-2\sum_{k=1}^{\infty }(-1)^{k-1}\frac{x}{k^2+x^2}=\int_{0}^{1 }\frac{t^{-ix}+t^{ix-1}}{1+t}dt$$

• Please do not write everything in one big formula. Break it up, and write text outside of formulas.
– quid
Aug 18, 2019 at 13:54

Integrate the function $$f(z) = \frac{z e^{iz}}{\cosh (z) + \cos(z)} \,$$ around the contour $$[-\sqrt{2} \pi N, \sqrt{2} \pi N] \cup \sqrt{2} \pi Ne^{i[0, \pi]}$$, where $$N$$ is a positive integer.

The integral vanishes on the semicircle as $$N \to \infty$$ because $$|f(z)|$$ decays exponentially fast to zero along the entire semicircle.

We therefore have

\begin{align} \int_{-\infty}^{\infty}f(x) \, \mathrm dx &= 2 \pi i \left(\sum_{n=0}^{\infty}\operatorname{Res} \left[f(z), \frac{(2n+1) \pi (1+i)}{2} \right] + \sum_{n=0}^{\infty}\operatorname{Res} \left[f(z), \frac{(2n+1) \pi (-1+i)}{2} \right]\right) \\ &= 2 \pi i \left(\sum_{n=0}^{\infty} \lim_{z \to \frac{(2n+1) \pi (1+i)}{2} }\frac{ze^{iz}}{\sinh(z) - \sin(z)} + \sum_{n=0}^{\infty} \lim_{z \to \frac{(2n+1) \pi (-1+i)}{2} }\frac{ze^{iz}}{\sinh(z) - \sin(z)} \right) \\ &= 2 \pi i \left(\frac{\pi}{2}\sum_{n=0}^{\infty} \frac{(2n+1)e^{-(2n+1) \pi /2}}{\cosh \left(\frac{(2n+1) \pi }{2} \right)} + \frac{\pi}{2}\sum_{n=0}^{\infty} \frac{(2n+1)e^{-(2n+1) \pi /2}}{\cosh \left(\frac{(2n+1) \pi }{2} \right)}\right) \\ &= 2 \pi^{2} i \sum_{n=0}^{\infty} \frac{(2n+1) e^{- (2n+1) \pi /2}}{\cosh \left(\frac{(2n+1)\pi }{2} \right)} \\ &= 4 \pi^{2}i \sum_{n=0}^{\infty} \frac{2n+1}{e^{(2n+1)\pi }+1}. \end{align}

Equating the imaginary parts on both sides of the equation, we get \begin{align} \sum_{n=0}^{\infty} \frac{2n+1}{e^{(2n+1) \pi }+1} &= \frac{1}{4 \pi^{2}} \int_{-\infty}^{\infty} f(x) \, \mathrm dx \\ &= \frac{1}{2 \pi^{2}}\int_{0}^{\infty} \frac{x \sin(x)}{\cosh (x) + \cos(x)} \, \mathrm dx \\ &= \frac{1}{\pi^{2}} \, \Im \int_{0}^{\infty} x \sum_{n=1}^{\infty} (-1)^{n-1} e^{(-1+i)nx} \, \mathrm dx \\ &= \frac{1}{\pi^{2}} \, \Im \sum_{n=1}^{\infty} (-1)^{n-1} \int_{0}^{\infty} x e^{(-1+i)nx} \, \mathrm dx \\ &= \frac{1}{\pi^{2}} \, \Im \sum_{n=1}^{\infty} (-1)^{n-1} \frac{1}{(1-i)^{2}n^{2}} \\ &= \frac{1}{2 \pi^{2}} \sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n^{2}} \\ &= \frac{1}{24} . \end{align}

In general, integrating the function $$g(z) = \frac{z^{4k+1} e^{iz}}{\cosh (z) + \cos(z)} \, , \quad k \in \mathbb{N}_{\ge 0},$$ around the same contour shows that

\begin{align} \sum_{n=0}^{\infty} \frac{(2n+1)^{4k+1}}{e^{(2n+1)\pi}+1} &= \frac{(-1)^{k}2^{2k-1}}{\pi^{4k+2}} \int_{0}^{\infty} \frac{x^{4k+1} \sin(x)}{\cosh(x) + \cos(x)} \, \mathrm dx \\ &= \frac{(-1)^{k}2^{2k-1}}{\pi^{4k+2}} \frac{1}{2^{2k}} (-1)^{k} (4k+1)! \sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n^{4k+2}} \\ &\overset{(1)}{=} \frac{B_{4k+2}}{2 (4k+2)} \left(2^{4k+1}-1\right), \end{align} where $$B_{k}$$ is the $$k$$th Bernoulli number.

$$(1)$$ https://en.wikipedia.org/wiki/Dirichlet_eta_function#Particular_values