Questions tagged [harmonic-numbers]

For questions regarding harmonic numbers, which are partial sums of the harmonic series. The $N$-th harmonic number is the sum of reciprocals of the first $N$ natural numbers.

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3
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2answers
152 views

Closed form of the sum $s_4 = \sum_{n=1}^{\infty}(-1)^n \frac{H_{n}}{(2n+1)^4}$

I am interested to know if the following sum has a closed form $$s_4 = \sum_{n=1}^{\infty}(-1)^n \frac{H_{n}}{(2n+1)^4}\tag{1}$$ I stumbled on this question while studying a very useful book about ...
7
votes
3answers
159 views

Compute the double sum $\sum_{n, m>0, n \neq m} \frac{1}{n\left(m^{2}-n^{2}\right)}=\frac{3}{4} \zeta(3)$

I am trying to compute the following double sum $$\boxed{\sum_{n, m>0, n \neq m} \frac{1}{n\left(m^{2}-n^{2}\right)}=\frac{3}{4} \zeta(3)}$$ I proceeded as following $$\sum_{n=1}^{\infty}\sum_{m=1}^...
2
votes
2answers
236 views

How to compute $\sum_{n=1}^\infty \frac{(-1)^n}{n^2}H_{2n}$

Edit In this post I computed the following integral $$\int_{0}^{1}\frac{\log(1-x)\log(1-x^2)}{x}dx=\frac{11}{8}\zeta(3)$$ Now I am trying to compute $$\boxed{\int_{0}^{1}\frac{\log(1-x)\log(1-x^4)}{x}...
4
votes
1answer
117 views

Where is my mistake $\int_{0}^{1}\frac{\log(1-x)\log(1-x^2)}{x}dx=\frac{7}{4}\zeta(3)$

Edit As commented bellow by @Donald Splutterwit and @ Elliot Yu, it seems that my computation is numerically correct and the post is wrong! I also added a corollary from this computation. I saw the ...
3
votes
0answers
125 views

Evaluating the integral: $\int _0^1\frac{\ln \left(x\right)\ln ^2\left(\frac{1-x}{1+x}\right)\operatorname{Li}_2\left(x\right)}{x}\:dx$

I'd like to evaluate and find the closed form for $$\int _0^1\frac{\ln \left(x\right)\ln ^2\left(\frac{1-x}{1+x}\right)\operatorname{Li}_2\left(x\right)}{x}\:dx$$ But it's very difficult since its ...
0
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1answer
50 views

Proving $\sum_{n=1}^{\infty}\dfrac{H^{(a)}_{n}}{n^b} = \zeta(a,b) + \zeta(a+b)$

I've come across the formula: $$\sum_{n=1}^{\infty}\dfrac{H^{(a)}_{n}}{n^b} = \zeta(a,b) + \zeta(a+b)$$ where $H^{(a)}_{n}$ is the Generalized Harmonic nuber and $\zeta(a,b)$ is the Multiple zeta ...
0
votes
1answer
59 views

Connection between $J_k= \sum_{n=1}^k e^{-n}=\frac{1-e^{-k}}{e-1} $ and $ f(x)=\sum_{n=1}^\infty e^{-n^x} ?$

Consider the geometric series$$J_k= \sum_{n=1}^k e^{-n}=\frac{1-e^{-k}}{e-1} $$ I'm wondering if this has any connection with: $$ f(x)=\sum_{n=1}^\infty e^{-n^x}. $$ $J_k$ can be interpreted as adding ...
7
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1answer
120 views

Harmonic numbers as ratio of two Determinants

Provide a proof to this interesting identity: $$\frac{\begin{vmatrix} 1^0 & 1^2 & 1^3 & \cdots & 1^n \\ 2^0 & 2^2 & 2^3 & \cdots & 2^n \\ \vdots & \vdots & \...
-1
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2answers
109 views

Value of sum of harmonic series

Let: $$ 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} + ... = A $$ Then: $$ \left(1 + \frac{1}{3} + \frac{1}{5} + ...\right) + \left(\frac{1}{2} + \frac{1}{4} + \frac{1}{6} + ...\right) = ...
0
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1answer
34 views

Calculating Harmonic Numbers

I'm hoping someone can confirm if I did this right, is the expression for calculating the nth harmonic number as written below? $$ H_n=\gamma+\lim_{h\to\infty}\left(\ln\left(h\right)-\sum_{k=n+1}^{h}\...
1
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1answer
34 views

How to check if number is harmonic divisor or not

I need to write a program which checks if the given number is harmonic divisor or not. So I search on the internet and found a definition of it but couldn't understand really what is harmonic divisor. ...
0
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0answers
34 views

Simplify a summation to the harmonic series

I've been given the following as a solution to a problem I'm working on, but I do not understand how to jump from step 1 to step 2. Kindly help me formulate an intermediate step between (1) and (2) to ...
2
votes
1answer
87 views

Sum of a p-series with coefficients

I am trying t find an expression for the partial sum of a p-like-series. The problem comes from trying to sum the elements of a matrix whose entries are inversely related to their distance from the ...
1
vote
1answer
63 views

Evaluating $\sum_{k=1}^{a}\frac{-1-H_k}{k(1-e)^k}$

Question : My attempt: Let $a=17399172$ $$\begin{align} &\sum_{k=1}^{a}\frac{-1-H_k}{\log_2\left(\sum_{j=0}^{k}\left(\ln\left(e^{C_j^k}\right)\right)\right)(1-e)^k} \\ &= \sum_{k=1}^{a}\frac{-...
5
votes
4answers
152 views

Closed form of $\sum_{m=1}^{\infty} \frac{(-1)^mH_{\frac{3m}{4}}}{3m}$

I've been working on an integral, namely: $${\displaystyle \int_0^1 \frac{x^2}{1 + x^3}\ln(1 - x^4)dx}$$ Which I managed to narrow down to the following expression: $$\sum_{m=1}^{\infty} \frac{(-1)^...
2
votes
2answers
61 views

Expected number of elements before 1 out of 2 hash tables are full?

From a previous asked question it became clear to me that the expected number of elements to fill up one hash table (assuming ideal randomness) is just the Harmonic number: $mH_m$ where $m$ is the ...
0
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0answers
20 views

Single instance of a harmonic series being less than the harmonic series multiplied by a constant such that C < 1

Problem This was posted last year as a problem of the week for my university without a solution. I cannot think of a single instance where this statement is true, and I don't know how to go about ...
1
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1answer
49 views

Bound on Truncated Alternating Harmonic Series

I'm trying to prove the following inequality $$\sum _{n=1}^{2N}\frac{(-1)^n}{n}+\log (2)<\frac{1}{4N+1}.$$ Most of the traditional inequalities I've seen for harmonic numbers aren't tight enough to ...
6
votes
2answers
137 views

Doubt in handling double sums of the type: $\sum_{n,k\in\mathbb{N}_1}\frac{1}{n^2k^2(n+k)}$

I'm having a little trouble calculating the double series given by the proposition below: $$\int_0^1\frac{\left[\text{Li}_2(x)\right]^2}{x}\mathrm dx=\sum_{n=1}^\infty\sum_{k=1}^\infty\frac{1}{n^2k^2(...
11
votes
2answers
292 views

Calculate $\sum_{k=1}^{\infty}\frac{|\sin(k x)|}{1+k^2}$

In https://math.stackexchange.com/a/4015346/198592 it was shown that the sum $$s(x) = \sum_{k=1}^{\infty}\frac{\sin(k x)}{1+k^2}$$ is exactly expressible in terms of the hypergeometric function. I ...
5
votes
1answer
108 views

Power series of $\log(1+x)\log(1-x)$

On the bottom of page 7 of the paper Logarithmic Integrals by Morshed (arXiv link) there is an "interesting generating function": $$ \log(1+x) \log(1-x) = \sum_{n=1}^\infty \left(H_n - H_{2n}...
8
votes
2answers
109 views

The expectation of $e^X \left(1-(1-e^{-X}\right)^n)$ when $X$ has Exponential Distribution

To my surprise, I was able to evaluate the following expression in Mathematica: $$E\left[e^X \left(1-(1-e^{-X}\right)^n) \right] = \frac{y}{y-1} \left(1-\frac{1}{\binom{n+y-1}{y-1}}\right)\quad X\sim\...
0
votes
0answers
35 views

Product of Generalized Harmonic Numbers

I'm trying to sum two indipendent random variabile, in particular two zipf's law, mathematica say that the product of two generalized harmonic number $H_{p,s}$ and $H_{t,k}$, $H_{p,s}H_{t,k}$ is eqaul ...
1
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0answers
49 views

Bounds on harmonic prime product

So it is known that $$ \prod_{p\, \text{prime}}\frac{p}{p-1} = \sum_{k=1}^\infty \frac{1}{k}. $$ It is also known that for any particular harmonic number $H_n$, that we can bound it by $$ H_n = \sum_{...
5
votes
1answer
237 views

infinite sum of harmonic number powers $\sum_{n=0}^{\infty}x^{H_n}$

I am looking for a simplified form of the infinite sum of harmonic exponentials $$ f(x) = \sum_{n=0}^{\infty}x^{H_n} = 1 + x + x^{3/2} + x^{11/6} + x^{25/12} + x^{137/60} + x^{49/20} + \ldots\\ =1+x\...
1
vote
1answer
33 views

Can we identify the pole of the Gamma function with the limit of the harmonic numbers?

The expansion of the gamma function around $x=0$ is $$\Gamma(x)=\frac{1}{x}-\gamma+O(x).$$ The Euler constant $\gamma$ is defined by $$\gamma=\lim_{n\to\infty}(H_n-\log n)$$ where $H_n$ is the $n$th ...
1
vote
1answer
70 views

Proving an equality involving harmonic numbers

I am working through the book (Almost) Impossible Integrals, Sums, and Series, and I am on the following problem: Let $n$ be a positive integer. Prove that the following equality holds. $$K_n = \...
0
votes
1answer
55 views

harmonic sum and digamma function [closed]

Let's consider digamma function $\Psi(x)$, i.e., $$ \Psi(x)=\frac{d}{dx}\ln \Gamma(x)=\dfrac{\Gamma\,'(x)}{\Gamma(x)}. $$ Prove the following $$ 1+\frac{1}{2}+\cdots+\frac{1}{n}=\gamma+\frac{1}{n}+\...
1
vote
1answer
144 views

How to Evaluate $ \sum_{n=1}^{\infty} \frac{(-1)^n}{n}H_{4n-3} $

I came across the following result: $$ \sum_{n=1}^{\infty} \frac{(-1)^n}{n}H_{4n-3} = \frac{\pi}{4} + \frac{\pi}{\sqrt{2}}-\frac{5\pi^2}{32}+\frac{\ln^2(2)}{8}-\frac{3\ln(2)}{2}+\frac{\ln^2(1+\sqrt{2})...
4
votes
1answer
127 views

Calculate $\int_0^1 \left\lfloor \frac{2020}{x} \right\rfloor - 2020\left\lfloor\frac1x\right\rfloor\,dx$

Calculate the definite integral $$\int_0^1 \left\lfloor \frac{2020}{x} \right\rfloor - 2020\left\lfloor\frac1x\right\rfloor\,dx$$ My Attempt: Let's make a start. Write $[2020/x]=n\ge 2020$ then ${{...
4
votes
2answers
156 views

Relationship between Partial Harmonic Sum and Logarithm.

Let $H_n$ denote the partial harmonic sum $$H_n = \sum_{j=1}^n \frac{1}{j} \ \ .$$ I saw in some lecture notes that $H_n \approx \ln(n)$ without any explanation given. I tried to understand this ...
1
vote
0answers
39 views

Series for inverse of Euler's harmonic number integral $H_x=\int_0^1\frac{1-t^x}{1-t}dt$

I'm looking to find a series for the inverse of Euler's analytic continuation of the harmonic numbers $$H_x=\int_0^1\frac{1-t^x}{1-t}dt$$ From what I've seen, this integral is equal to the power ...
4
votes
2answers
193 views

Closed form of the sum $\sum_{n=1}^{\infty}\frac{H_n}{n^x}$

Some days ago I derived the identity $$\sum_{n=1}^{\infty}\frac{H_n}{n^2}=2\zeta(3)$$ where $H_n$ is the $n$th Harmonic number. Other related identities include $$\sum_{n=1}^{\infty}\frac{H_n}{n^3}=\...
2
votes
1answer
113 views

Another bizarre sum involving a binomial coefficient and inverse powers of integers.

In the attempt to answer Binomial identity involving Harmonic numbers we stumbled on the following problem. Let $i\ge 0 $ and $k \ge i+2 $ and $p \ge 1$ be integers. Consider a following sum: \begin{...
0
votes
1answer
48 views

Divergence of Harmonic Series

I understand that the divergence of the harmonic series is a classic proof but what I don't understand is the way it seems to contradict standard methods of finding if a series converges or not. For ...
3
votes
1answer
168 views

Binomial identity involving Harmonic numbers

Using numeric experiments, I have found the following identity $$ k! H_k+\sum _{j=1}^k \frac{\sum _{i=j}^k (-1)^i \binom{k}{i} \left((j-i)^k-(-i-n+1)^k\right)}{j+n-1}=0 $$ where $H_k = \sum_{i=1}^k 1/...
1
vote
2answers
49 views

Inductive proof that sum of reciprocals of odd natural numbers diverges?

I am trying to prove via induction that for any natural number $N$, there exists an $n$ such that: $$ \sum_{i=1}^{n} \dfrac{1}{2i-1} > N $$ Or $$ 1 + \dfrac{1}{3} + \dfrac{1}{5} + ... + \dfrac{1}{...
2
votes
2answers
71 views

Inequality of harmonic number $\left| \sum_{i=1}^{n}\frac{1}{i}-\log{n}-\gamma \right| \leq \frac{10}{n}$

In my Number Theory textbook, it was quoted without proof that for all positive integers $n$, $$\left| \sum_{i=1}^{n}\frac{1}{i}-\log{n}-\gamma \right| \leq \frac{10}{n}$$ where $\gamma = 0.577...$ is ...
3
votes
1answer
414 views

How to evaluate $\sum _{n=1}^{\infty }\left(\frac{H_n^2+H_n^{\left(2\right)}}{n}\right)^2$ in a particular way.

How to evaluate: $$\sum _{n=1}^{\infty }\left(\frac{H_n^2+H_n^{\left(2\right)}}{n}\right)^2,$$ without splitting the expression into more sums. Here $H_n^{\left(m\right)}=\sum _{k=1}^n\frac{1}{k^m}$ ...
1
vote
0answers
35 views

A sum with harmonic number

i trying to evaluate this and i made some results , so could any one try this: $ \displaystyle \sum_{k=1}^{n} (\frac{{\rm H}_{k}}{n+1-k}) $.
2
votes
1answer
51 views

Simplify the following sum $\sum_{i=1}^n\frac1{n-(i-1)}$

Extremely simple question, but one I am struggling with (I haven't taken a math class for a couple years so this may be very easy). I just need to simplify the following sum, but can't seem to figure ...
-1
votes
1answer
41 views

Two ways to calculate the same sum where the harmonic number pops up.

Can someone help me with this equality? Prove that $$\sum_{i=1}^n \frac{x^i}{i} = \sum_{i=1}^n {n \choose i}\frac{(x-1)^i}{i} + H_n$$ where $H_n$ is the harmonic number.
1
vote
1answer
76 views

Prove that for any integer $n > 3$ there exist positive integers $a_1 > \dots > a_n$ such that $1 = \frac{1}{a_1} + \dots + \frac{1}{a_n}$ [duplicate]

I have tried to approach this problem in a variety of ways, like triyng to construct a solution, looking at telescoping series, geometrically... But I have been unable to solve it. I would appreciate ...
3
votes
5answers
161 views

Solve the recurrence relation: $na_n = (n-4)a_{n-1} + 12n H_n$

I want to solve $$ na_n = (n-4)a_{n-1} + 12n H_n,\quad n\geq 5,\quad a_0=a_1=a_2=a_3=a_4=0. $$ Does anyone have an idea, what could be substituted for $a_n$ to get an expression, which one could just ...
1
vote
2answers
106 views

Approximation of Sum of Partial Harmonic series with only odd terms with elementary methods

Let $a=\frac{1^2}{1}+\frac{2^2}{3}+\frac{3^2}{5}…+\frac{1001^2}{2001}$ and $b=\frac{1^2}{3}+\frac{2^2}{5}+\frac{3^2}{7}…+\frac{1001^2}{2003}$ Find the closest integer to $a-b$ Using the identity $a^2-...
6
votes
1answer
167 views

Proof of an exact formula for $H_n$

The $n$th harmonic number $H_n$ is defined as $$H_n=\sum_{n\geq k\geq 1}\frac{1}{k}$$ A good approximation for this is $$H_n\approx \gamma+\log n +\frac{1}{2n}$$ Where $\gamma$ is the Euler-Mascheroni ...
13
votes
2answers
673 views

Evaluating the challenging sum $\sum _{k=1}^{\infty }\frac{H_{2k}}{k^3\:4^k}\binom{2k}{k}$.

I managed to evaluate the sum, my approach can be found $\underline{\operatorname{below as an answer}}$, I'd truly appreciate if any of you could share new methods to evaluate this series, thank you. ...
0
votes
1answer
54 views

Tight upper bound on $\sum_{j=1}^{n}(-1)^{j-1}{n \choose j}\frac{1}{j}$?

I would like to prove the following inequality (upper bound): \begin{align} \xi(n) = \sum_{j=1}^{n}(-1)^{j-1}{n \choose j}\frac{1}{j} \leq 3\ln(n), \end{align} for $n\geq 2$. I have made an attempt to ...
1
vote
0answers
108 views

Calculate the sum of $\:\frac{x}{5} + \:\frac{x}{11} + \:\frac{x}{17} … + \:\frac{x}{p} $

Let's say $x = 161$, and I want to calculate the sum of all : $$\:\frac{x}{5} + \:\frac{x}{11} + \:\frac{x}{17} .... + \:\frac{x}{p} $$ $$(\text{pattern} = \:\frac{x}{\left(6n-1\right)}\:) $$ but the ...
8
votes
2answers
452 views

Evaluating $\sum _{k=1}^{\infty }\frac{H_k}{4^k\left(2k+1\right)}\binom{2k}{k}$.

My attempt. $$\sum _{k=1}^{\infty }\frac{H_k}{4^k\left(2k+1\right)}\binom{2k}{k}$$ $$=\frac{1}{2}\sum _{k=1}^{\infty }\frac{H_k}{k\:4^k}\binom{2k}{k}-\frac{1}{2}\sum _{k=1}^{\infty }\frac{H_k}{k\:4^k\...

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