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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Symmetry relation for finite sums of generalized harmonic numbers

EDITs - extension to alternating sums (6) - extension to general sums with parameter $x$ (8) - extension to general sums with two parameters (11) Extended post The relation $$\sum _{k=1}^{n } \left(...
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Sum, series and Harmonic numbers [duplicate]

I found the solution of series on Wolfram Alpha http://www.wolframalpha.com/input/?i=sum+1%2F(2k%2B1)%2F(2k%2B2)+from+1+to+n%2F2 for $\alpha = 1$ $\sum\limits_{k=1}^{n/2} \left(\frac{1}{\left(2k-1+2^...
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On twisted Euler sums

An interesting investigation started here and it showed that $$ \sum_{k\geq 1}\left(\zeta(m)-H_{k}^{(m)}\right)^2 $$ has a closed form in terms of values of the Riemann $\zeta$ function for any ...
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Finding the general/closed form of $\sum_{k=1}^n k^a$ [duplicate]

I recently noticed the following: $$\sum_{i=1}^ni=\frac{n(n+1)}{2}$$ $$\sum_{i=1}^{n}i^2 = \dfrac{n(n+1)(2n+1)}{6} = \Bigg(\sum_{i=1}^ni\Bigg) \cdot \frac{2n+1}{3}$$ $$\sum\limits_{i = 1}^n i^3 = \...
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For given $k,N\in\mathbb{N}$, how to compute $\sum_{i=0}^Ni^k$? [duplicate]

For given $k$ and $N$, $k,N\in\mathbb{N}$, how to compute $\sum_{i=0}^Ni^k$? We have: $\sum^N_{i=0}i=\frac{N(N+1)}2$ Also according to what I found in the Internet we have $\sum^N_{i=0}i^2=\frac{n(n+...
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Why can we maximize $\frac{k-1}{n}(\mathrm{H}_{n-1}-\mathrm{H}_{k-2})$ for $2\leq k\leq n-1$ by substituting $\mathrm{H}_l$ with $\ln(l)$?

During an excercise session in a basic course of probability it was shown that the secretary problem can be reduced to solving the following task: For a given natural $n$ optimize $2\leq k\leq n-1$ so ...
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69 views

Partial sum involving harmonic numbers

QUESTION: I need to know how to compute the partial sum $$\sum_{k=1}^n \frac{H_{k+1}^2-H_{k+1}^{(2)}}{k+2}$$ in terms of the generalized harmonic numbers $H_n^{(m)}$. CONTEXT: This problem arose ...
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Generating function for harmonic number times log ($H_k log(k)$)

During my studies of generating functions for expressions composed of harmonic numbers and its asyptotic approximations I found a simple sum which seems to have been neglected so far $$s171219a=\sum ...
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A closed form of the family of series $\sum _{k=1}^{\infty } \frac{\left(H_k\right){}^m-(\log (k)+\gamma )^m}{k}$ for $m\ge 1$

Introduction Inspired by the work of Olivier Oloa [1] and the question of Vladimir Reshetnikov in a comment I succeeded in calculating the closed form of the sum $$s_m = \sum _{k=1}^{\infty } \frac{\...
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Prove series form of fractional harmonic numbers

Let $H_\alpha$ be the $\alpha$th fractional harmonic number so that $$ H_\alpha = \int_0^1 \frac{1-x^\alpha}{1-x}\,\text dx. $$ I want to directly show $$ H_\alpha = \sum_{k=1}^\infty \frac{\alpha}{k(...
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How can we prove $\sum\limits_{n=1}^{k}\frac{f(m,n)}{n}\approx\frac{\varphi(m)}{m}(\ln(k\prod\limits_{p|m}p^{\frac{1}{p-1}})+\gamma)$?

First we need to create a function $$f(m,n)=\begin{cases} 1&\text{if $m,n$ - coprime}\\ 0&\text{otherwise}\\ \end{cases}$$ Then we can be sure, that for large $k$ and $|\mu(m)|=1$ $$\sum\...
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Sums and harmonic series

I found the solution of series on Wolfram Alpha http://www.wolframalpha.com/input/?i=sum+1%2F(2k%2B1)%2F(2k%2B2)+from+1+to+n $ \sum\limits_{k=1}^{n} \left(\frac{1}{2k+1} - \frac{1}{2k+2}\right) = \...
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Harmonic Number for $H_{n+1}$ and $H_{n+\frac{1}{2}}$

According to the definition of harmonic number $H_n = \sum\limits_{k=1}^n\frac{1}{k}$. How we can define $H_{n+1}$ and $H_{n+\frac{1}{2}}$?
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Prove that: $\frac{1}{n}+\frac{1}{n+1}+\cdots+\frac{1}{2n}\ge\frac{2}{3}$

I've got three inequalities: $\forall n\in\mathbb N:$ $$\frac{1}{n}+\frac{1}{n+1}+\cdots+\frac{1}{2n} \ge\frac{1}{2}$$ $$\frac{1}{n}+\frac{1}{n+1}+\cdots+\frac{1}{2n} \ge\frac{7}{12}$$ $$\frac{1}{n}...
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3answers
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Closed expressions for harmonic-like multiple sums

Inspired by Find the sum of the double series $\sum_{k=1}^\infty \sum_{j=1}^\infty \frac{1}{(k+1)(j+1)(k+1+j)} $ here are some related problems. Prove that $$w(2,1) = \sum _{i=1}^{\infty } \sum _{j=...
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Limit Involving Generalised Harmonic Numbers

Is there an easy/straight-forward way to evaluate the limit $$ \lim_{n\to\infty} \frac{\pi^4-90H_n^{(4)}}{4\pi^2(\pi^2-6H_n^{(2)})},$$ where $H_{n}^{(r)}$ denotes the $n$th generalised Harmonic number ...
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On the sums $\sum_{i=1}^{n} \sum_{j=0}^{m-1} \dfrac{a_j}{mi-j}$ as $n \to \infty$

This was inspired by Evaluating $\sum_{k=0}^\infty \left(\frac{1}{5k+1} - \frac{1}{5k+2} - \frac{1}{5k+3} + \frac{1}{5k+4} \right)$ Let $s(n) =\sum_{i=1}^{n} \sum_{j=0}^{m-1} \dfrac{a_j}{mi-j} $ and $...
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Another “closed form expression” for a generating function involving harmonic numbers.

This question is closely related to Calculating alternating Euler sums of odd powers. Let $p\ge 1$ and $q\ge 1$ be integers and $1> x\ge 0$ be real. We use the following definition: \begin{...
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Find the sum of the double series $\sum_{k=1}^\infty \sum_{j=1}^\infty \frac{1}{(k+1)(j+1)(k+1+j)} $

First, the original problem follows, $$\sum_{k=1}^\infty \frac{H_{k+1}}{k(k+1)}$$ where $$H_{k}=\sum_{j=1}^k \frac{1}{j}$$ is the $k$-th partial sum of harmonic series. Using the following identity, ...
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Set Consisting of Harmonics of Integer Powers of 2?

If one defines "harmonics" as natural number fractions of a number, is there a name for the set consisting of all harmonics of all integer powers of two?
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Closed form expressions for harmonic sums

It is well known that there are deep connections between harmonic sums (discrete infinite sums that involve generalized harmonic numbers) and poly-logarithms. Bearing this in mind we have calculated ...
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2answers
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I can't figure out my mistake in calculating the harmonic series

Let's say $a = H_{\infty}$, so $a = \sum_{k=1}^\infty \frac{1}{k}$. $$a = 1 + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{\infty}$$ Now if we take $a - a + 1$ we get $$a - a + 1 = 1 + \frac{1}{2} + ...
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Harmonic number identity $\sum_{k=1}^{n}H_k=(n+1)(H_{n+1}-1)$

Prove using generating functions that for Harmonic numbers $H_n= \sum_{j=1}^{n}\frac{1}{j}$ $$\sum_{k=1}^{n}H_k=(n+1)(H_{n+1}-1) (*)$$ The generating function for harmonic numbers is $$\sum_{n \in \...
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Why there isn't any exact formula to calculate summation of n terms of a HP?

When I search over the net to find such a formula all I find is the approximations but not exact. So, is it even possible to find the exact formula or is it the case that we will never be able to find ...
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Series whose $a_n \rightarrow 0$ as $n \rightarrow \infty$ and diverges

I was wondering if there are series whose $a_n \rightarrow 0$ as $n \rightarrow \infty$ and diverges. I know one example, that is $a_n = \frac{1}{n}$. Slight modifications don't count, such as $a_n = \...
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1answer
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About the partition of some reciprocals

Does there exists a partition of the reciprocals $\frac{1}{2}, \frac{1}{3},\cdots,\frac{1}{12}$, into two sets $A$ and $B$ such that $\displaystyle\sum_A\frac{1}{n} -\sum_B\frac{1}{n}=1$?
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Let $x_n$ denote the smallest integer such that $\sum\limits_{k=n}^{x_n} \frac 1k >1$, then $x_n\sim en$

For $n\geq 2$, let $x_n$ denote the smallest integer such that $\displaystyle \sum_{k=n}^{x_n} \frac 1k >1$. Prove that $\lim_n \frac{x_n}{n}\sim e$. I've found this problem in an old handout of ...
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A pair of composed integral transforms from Mellin and Laplace transforms

We can create a forward transform $T$ which is a composition of the inverse Mellin $\mathcal{M}^{-1}$ and Laplace $\mathcal{L}$ transforms $$ T[\phi] = \mathcal{L}\mathcal{M}^{-1}[\phi]=\mathcal{L}\...
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Simple proof of showing the Harmonic number $H_n = \Theta (\log n)$

Consider the partial sum of the divergent Harmonic series $H_n = \sum\limits_{k = 1}^{n}\frac{1}{k}$. I recently saw a question which required finding out the asymptotic bounds of $H_n$. Now, I could ...
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1answer
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Tricky partial rational sum

I'm looking for a simplification of $$ \sum _{k=r+1}^{2 r} \frac{2 k+2 r+1}{2 k^2-k (2 r+1)+2 r (r+1)}\:. $$ Mathematica gives a somewhat tautological result in terms of the digamma function $\psi$: $...
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How to evaluate the sum $\sum_{n=1}^{\infty}\frac{1}{(2n+1)(2n+2)}\left(1+\frac{1}{2}+…+\frac{1}{n}\right)$

How to evaluate the sum: $$S=\sum_{n=1}^{\infty}\frac{1}{(2n+1)(2n+2)}\left(1+\frac{1}{2}+...+\frac{1}{n}\right)$$ Can anyone help me,I really appreciate it.
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1answer
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Evaluating $~\sum\limits_{n>0}H(e^n)-n-\gamma~$ in Closed Form

Inspired by these three questions, I asked myself whether $$\sum_{n>0}\Big[~H(e^n)-n-\gamma~\Big]~=~0.278091975548622251874828828459627630\ldots$$ might also possesses a closed form expression, ...
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28 views

What is the minimum number of operations needed to calculate exactly the nth Harmonic number

A teacher once told me the minimum number of operations needed to calculate the exact value of the nth Harmonic number was n operations. Using something similar to Euler's proof of: $\sum_{i=1}^n i = \...
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480 views

A conjectured result for $\sum_{n=1}^\infty\frac{(-1)^n\,H_{n/5}}n$

Let $H_q$ denote harmonic numbers (generalized to a non-integer index $q$): $$H_q=\sum_{k=1}^\infty\left(\frac1k-\frac1{k+q}\right)=\int_0^1\frac{1-x^q}{1-x}dx=\gamma+\psi(q+1),\tag1$$ where $\psi(z)=\...
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Infinite sum of harmonic number over general polynomial

As an extension to my question Infinite sum of harmonic number over polynomial of 2nd degreee I found a general formula for a polynomial of arbitrary degree which I had presented as a self answer but ...
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2answers
82 views

Infinite sum of harmonic number over polynomial of 2nd degreee

Inspired by the recent problem How to evaluate the sum $\sum_{n=1}^{\infty}\frac{1}{(2n+1)(2n+2)}\left(1+\frac{1}{2}+...+\frac{1}{n}\right)$ I asked for a generalization, and I found it with ...
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5answers
572 views

Prove that $ 1 + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{n} = \mathcal{O}(\log(n)) $.

Prove that $ 1 + \dfrac{1}{2} + \dfrac{1}{3} + \cdots + \dfrac{1}{n} = \mathcal{O}(\log(n)) $, with induction. I get the intuition behind this question. Clearly, the given function isn’t even growing ...
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592 views

Identities For Generalized Harmonic Number

I have been searching for identities involving generalized harmonic numbers \begin{equation*}H_n^{(p)}=\sum_{k=1}^{n}\frac{1}{k^p}\end{equation*} I found several identities in terms of $H_n^{(1)}$, ...
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Formulas involving the polynomials $\frac{1}{n!}x^n(a_n-b_nx)^n,$ where the coprime integers $a_n,b_n$ satisfy $H_n=\frac{a_n}{b_n}$ for $n\geq 1$

For integers $n\geq 1$ let $$H_n=1+\frac{1}{2}+\ldots+\frac{1}{n}$$ the $nth$ harmonic number. After I've seen the form of the polynomials used by Niven in [1] I wanted to create a puzzle with a new ...
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Exploring the constant $-\int_0^1\log\left(1-H_x\right)\log\left(1+H_{2x}\right)\,dx$, where $H_y$ are harmonic numbers

This morning I try to create interesting integrals involving harmonic numbers. See this Wikipedia. And look at, also if you need it, the definition of the Harmonic number $H_x$ using the digamma ...
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Comparison of sum and integral over squares of Harmonic numbers

This is an extension of the simpler question [1] This time we compare sum and integral over the squares of the harmonic numbers (see [1] for definitions) The sum is $$f_{s2}(n) = \sum_{k=0}^n H_k^2$...
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Comparison of sum and integral over Harmonic number

The sum of Harmonic numbers $H_n = \sum_{k=1}^n \frac{1}{n}, H_0 = 0$ defined by $$f_s(n) = \sum_{k=0}^n H_k$$ is given by $$f_s(n) = (n+1) H_n - n$$ Now define the integral $$f_i(n) = \int_0^n ...
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2answers
274 views

Is there any closed form for the finite sum $1+\frac12+\frac13+\frac14+\dots+\frac1n?$ [duplicate]

I know that the infinite summation $$1+\dfrac{1}{2}+\dfrac{1}{3}+\dfrac{1}{4}+...+\dfrac{1}{n}+...$$ is divergent and also the sequence $$1+\dfrac{1}{2}+\dfrac{1}{3}+\dfrac{1}{4}+...+\dfrac{1}{n}-\ln ...
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2answers
74 views

Is the function $\frac{\operatorname{Harmonic}(x)}{\operatorname{Airy}(x)}$ a convex function for real numbers $0<x<1$?

For real numbers $0\leq x\leq 1$, I know how to exlore some aspects of the graph ( I was playing with Wolfram Alpha online calculator about its plot, see next paragraph) of the function $$f(x)=\frac{...
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1answer
114 views

Infinite sum involving complete and incomplete zeta function

While playing around with complete and incomplete zeta functions I found a nice formula which to my knowledge was not discussed in MSE. Here it is $$\sum_{k=2}^\infty (\zeta(k)-1) = 1\tag{1}$$ and ...
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5answers
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A series involving the harmonic numbers : $\sum_{n=1}^{\infty}\frac{H_n}{n^3}$

Let $H_{n}$ be the nth harmonic number defined by $ H_{n} := \sum_{k=1}^{n} \frac{1}{k}$. How would you prove that $$\sum_{n=1}^{\infty}\frac{H_n}{n^3}=\frac{\pi^4}{72}?$$ Simply replacing $H_{n}$ ...
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0answers
48 views

Partial sum formula for $\sum_{x=1}^n \frac{1}{bx - x^2}$ at specific limit without using digamma function

Given the following sum: $\sum_{x=1}^n \frac{1}{bx - x^2}$ (b is an integer constant) It appears to me that partial sum cannot be calculated without using Harmonic numbers, or Digamma function. ...
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1answer
324 views

Is there a closed form for $\sum _{k=1}^n \frac{1}{k}H_{k-1}^2$?

Introduction This question appeared in the study (Sum of powers of Harmonic Numbers) of the sums of fourth powers of harmonic numbers $H_n = \sum _{k=1}^n \frac{1}{k}$ for $n=1,2,3,...$ and $H_0 = 0$....
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136 views

Is this sum $\sum_{k=0}^{n}{n\choose k}\sum_{i=0}^{k}(-1)^i{k\choose i}H_{n+k-i}=H_{2n}$ correct?

Given the double sum, which has a closed form of $H_{2n}$ $$\sum_{k=0}^{n}{n\choose k}\sum_{i=0}^{k}(-1)^i{k\choose i}H_{n+k-i}\tag1$$ Where $H_n$ is the Harmonic number $(1)$ is quite an ...
3
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1answer
154 views

Fibonacci and Harmonic

Here is a cute series that I came across and cannot seem to tackle at all. Question: Calculate $$\mathcal{S} = \sum_{n=1}^{\infty} \frac{F_n \mathcal{H}_{n-1}^{(2)}}{n^2 \binom{2n}{n}}$$ where $F_n$...