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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.

124
votes
22answers
8k views

Why does the series $\sum_{n=1}^\infty\frac1n$ not converge?

Can someone give a simple explanation as to why the harmonic series $$\sum_{n=1}^\infty\frac1n=\frac 1 1 + \frac 12 + \frac 13 + \cdots $$ doesn't converge, on the other hand it grows very slowly?...
60
votes
12answers
5k views

The limit of truncated sums of harmonic series, $\lim\limits_{k\to\infty}\sum_{n=k+1}^{2k}{\frac{1}{n}}$

What is the sum of the 'second half' of the harmonic series? $$\lim\limits_{k\to\infty}\sum\limits_{n=k+1}^{2k}{\frac{1}{n}} =~ ?$$ More precisely, what is the limit of the above sequence of partial ...
178
votes
10answers
27k views

Is there an elementary proof that $\sum \limits_{k=1}^n \frac1k$ is never an integer?

If $n>1$ is an integer, then $\sum \limits_{k=1}^n \frac1k$ is not an integer. If you know Bertrand's Postulate, then you know there must be a prime $p$ between $n/2$ and $n$, so $\frac 1p$ ...
34
votes
5answers
11k views

How to find the sum of this series : $1+\frac{1}{2}+ \frac{1}{3}+\frac{1}{4}+\dots+\frac{1}{n}$

Problem : How to find the sum of this series : $1+\frac{1}{2}+ \frac{1}{3}+\frac{1}{4}+\dots+\frac{1}{n}$ This is a Harmonic progression : So is this formula correct to sum the series : $\frac{(...
8
votes
4answers
575 views

Proving Binomial Identity without calculus

How to establish the following identities without the help of calculus: For positive integer $n, $ $$\sum_{1\le r\le n}\frac{(-1)^{r-1}\binom nr}r=\sum_{1\le r\le n}\frac1r $$ and $$\sum_{0\le r\le ...
15
votes
6answers
20k views

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 ...
12
votes
7answers
26k views

Summing Finitely Many Terms of Harmonic Series: $\sum_{k=a}^{b} \frac{1}{k}$

How do I calculate sum of a finite harmonic series of the form : $$\sum_{k=a}^{b} \frac{1}{k} = \frac{1}{a} + \frac{1}{a+1} + \frac{1}{a+2} +\cdots \frac{1}{b}.$$ Is there a general formula for this?...
25
votes
2answers
2k views

Showing that $\lim_{n\to\infty}\sum^n_{k=1}\frac{1}{k}-\ln(n)=0.5772\ldots$

How to show that $$\lim_{n\to\infty}\left[\sum^n_{k=1}\frac{1}{k}-\ln(n)\right]=0.5772\ldots$$ No clue at all. Need help! Appreciated!
43
votes
5answers
8k views

Do harmonic numbers have a “closed-form” expression?

One of the joys of high-school mathematics is summing a complicated series to get a “closed-form” expression. And of course many of us have tried summing the harmonic series $H_n =\sum \limits_{k \leq ...
30
votes
4answers
1k views

Infinite Series $\sum_{n=1}^\infty\frac{H_n}{n^22^n}$

How can I prove that $$\sum_{n=1}^{\infty}\frac{H_n}{n^2 2^n}=\zeta(3)-\frac{1}{2}\log(2)\zeta(2).$$ Can anyone help me please?
49
votes
7answers
8k views

Infinite Series $\sum_{n=1}^\infty\frac{H_n}{n^32^n}$

I'm trying to find a closed form for the following sum $$\sum_{n=1}^\infty\frac{H_n}{n^3\,2^n},$$ where $H_n=\displaystyle\sum_{k=1}^n\frac{1}{k}$ is a harmonic number. Could you help me with it?
11
votes
5answers
10k views

Proving $\frac{1}{n+1} + \frac{1}{n+2}+\cdots+\frac{1}{2n} > \frac{13}{24}$ for $n>1,n\in\Bbb N$ by Induction

Proving $\frac{1}{n+1} + \frac{1}{n+2}+\cdots+\frac{1}{2n} > \frac{13}{24}$ for $n>1,n\in\Bbb N$ To solve it I used induction but it is leading me nowhere my attempt was as follows: Lets ...
40
votes
7answers
3k views

Infinite Series $\sum\limits_{n=1}^\infty\left(\frac{H_n}n\right)^2$

How can I find a closed form for the following sum? $$\sum_{n=1}^{\infty}\left(\frac{H_n}{n}\right)^2$$ ($H_n=\sum_{k=1}^n\frac{1}{k}$).
43
votes
5answers
1k views

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 $H_{n}$ ...
19
votes
5answers
9k views

Is there any formula for the series $1 + \frac12 + \frac13 + \cdots + \frac 1 n = ?$

Is there any formula for this series? $$1 + \frac12 + \frac13 + \cdots + \frac 1 n .$$
15
votes
2answers
12k views

Simple proof Euler–Mascheroni $\gamma$ constant

I'm searching for a really simple and beautiful proof that the sequence $(u_n)_{n \in \mathbb{N}} = \sum\nolimits_{k=1}^n \frac{1}{k} - \log(n)$ converges. At first I want to know if my answer is OK. ...
2
votes
2answers
300 views

Find the limit $\lim_{n\rightarrow\infty}\sum_{k=1}^{n}\frac{1}{k+n}$ [duplicate]

$$\lim_{n\rightarrow\infty}\sum_{k=1}^{n}\frac{1}{k+n}$$ Could you please give me a hint how to find this limit?
21
votes
1answer
55k views

Is there a partial sum formula for the Harmonic Series? [duplicate]

There is a partial sum formula for $$\sum_{x=1}^n x^1 = \frac{n(n+1)}{2}$$ and even one when the exponent of $x$ is $0$: $$\sum_{x=1}^n x^0 = n$$ but I cannot find one for exponent $-1$: $$\sum_{x=1}^...
9
votes
4answers
3k views

Showing inequality for harmonic series.

I want to show that $$\log N<\sum_{n=1}^{N}\frac{1}{n}<1+\log N.$$ But I don't know how to show this.
11
votes
2answers
568 views

Sum of powers of Harmonic Numbers

This is a natural extension of the question Sum of Squares of Harmonic Numbers. I became interested in this question while studying the problem A closed form of $\sum_{n=1}^\infty\left[ H_n^2-\left(\...
28
votes
1answer
2k views

Sum of Squares of Harmonic Numbers

Let $H_n$ be the $n^{th}$ harmonic number, $$ H_n = \sum_{i=1}^{n} \frac{1}{i} $$ Question: Calculate the following $$\sum_{j=1}^{n} H_j^2.$$ I have attempted a generating function approach but ...
6
votes
1answer
574 views

Why is $e$ close to $H_8$, closer to $H_8\left(1+\frac{1}{80^2}\right)$ and even closer to $\gamma+\log\left(\frac{17}{2}\right) +\frac{1}{10^3}$?

The eighth harmonic number happens to be close to $e$. $$e\approx2.71(8)$$ $$H_8=\sum_{k=1}^8 \frac{1}{k}=\frac{761}{280}\approx2.71(7)$$ This leads to the almost-integer $$\frac{e}{H_8}\approx1....
2
votes
1answer
247 views

Re-Expressing the Digamma

I was reading some articles on the digamma function, and I was wondering if anyone knows how to express the digamma function $\psi^{(0)}(n)$ in terms of a trigonometric function or a logarithmic ...
7
votes
6answers
361 views

Find $\lim_{n\to\infty} (1+\frac{1}{2}+…+\frac{1}{n})\frac{1}{n}$

Find the following limit: $$\lim_{n\to\infty} \left(1+\frac{1}{2}+...+\frac{1}{n}\right)\frac{1}{n}$$ My intuition says that this goes to zero, because $1/n$ goes much faster to zero than the ...
25
votes
4answers
1k views

A series involves harmonic number

How do we get a closed form for $$\sum_{n=1}^\infty\frac{H_n}{(2n+1)^2}$$
3
votes
2answers
5k views

Asymptotic estimate of the sequence of harmonic series $\sum_{k=1}^{n} \frac{1}{k} $ [closed]

Asymptotic estimate of the sequence of partial sums, for $n\rightarrow \infty$: $$ s_n=\sum_{k=1}^{n} \frac{1}{k} $$
3
votes
4answers
295 views

Test for the convergence of the sequence $S_n =\frac1n \left(1 + \frac{1}{2} + \frac{1}{3} + \cdots+ \frac{1}{n}\right)$

$$S_n =\frac1n \left(1 + \frac{1}{2} + \frac{1}{3} + \cdots+ \frac{1}{n}\right)$$ Show the convergence of $S_n$ (the method of difference more preferably) I just began treating sequences in school, ...
2
votes
3answers
3k views

Convergence N'th Harmonic number minus the Natural Logarithm of N. [duplicate]

I was hoping if someone could show me the proof of exactly why this converges to the Euler–Mascheroni constant.
15
votes
2answers
413 views

Prove that neither $A$ nor $B$ is divisible by $5$

Let the sum $$ {1+ \frac12 + \frac13 + \frac 14+ \dots +\frac1{99} + \frac 1{100}}$$ be written as $\frac AB$, where $A$ and $B$ are positive integers with no common factors. Show that neither $A$ ...
13
votes
3answers
1k views

Harmonic Numbers series I

Can it be shown that \begin{align} \sum_{n=1}^{\infty} \binom{2n}{n} \ \frac{H_{n+1}}{n+1} \ \left(\frac{3}{16}\right)^{n} = \frac{5}{3} + \frac{8}{3} \ \ln 2 - \frac{8}{3} \ \ln 3 \end{align} where $...
14
votes
3answers
215 views

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.
14
votes
8answers
3k views

prove $\frac{1}{ n+1}+\frac{1}{ n+2}+\cdots+\frac{1}{2n}<\frac{25}{36}$ by mathematical induction

How to prove $$\frac{1}{ n+1}+\frac{1}{ n+2}+\cdots+\frac{1}{2n}<\frac{25}{36}$$ by Mathematical induction,n$\ge $1
5
votes
4answers
439 views

Prove this inequality: $\frac n2 \le \frac{1}{1}+\frac{1}{2}+\frac{1}{3}+…+\frac1{2^n - 1} \le n$

$\dfrac{n}{2} \le \dfrac{1}{1}+\dfrac{1}{2}+\dfrac{1}{3}+...+\dfrac{1}{2^n - 1} \le n $ I've Tried for hours but didn't got any striking idea. I don't have any efforts to show rather than induction. ...
10
votes
0answers
696 views

stirling numbers and harmonic number identities

Permit me a brief introduction before I state the question, three questions in fact. Inspired by this MSE link I computed the following harmonic sum identities: $$1/6\, \left( {H_{{n}}}^{(1)} \right) ...
6
votes
4answers
173 views

Prove that $1<\frac{1}{n+1}+\frac{1}{n+2}+…+\frac{1}{3n+1}$

Prove that $1<\dfrac{1}{n+1}+\dfrac{1}{n+2}+...+\dfrac{1}{3n+1}$. By using the Mathematical induction. Suppose the statement holds for $n=k$. Then for $n=k+1$. We have $\dfrac{1}{k+2}+\dfrac{1}{k+...
6
votes
2answers
220 views

Showing that $\sum_{i=1}^n \frac{1}{i} \geq \log{n}$

I have been trying to prove this by induction on $n\in \mathbb{N}$, but this approach seemed to get me nowhere. I have a suspicion it might be necessary to express $\log{n}$ as $\int_1^n 1/x\text{ }dx$...
3
votes
2answers
94 views

A proof of $\sum_{i=1}^{n}(-1)^{i-1}\binom{n}{i}\frac{1}{i}=\frac{1}{1}+\frac{1}{2}+…+\frac{1}{n}$ [duplicate]

How can I prove $$\sum_{i=1}^{n}(-1)^{i-1}\dbinom{n}{i}\frac{1}{i}=\frac{1}{1}+\frac{1}{2}+....+\frac{1}{n}\quad ?$$ I have tried induction, but it doesn't work.
1
vote
3answers
150 views

Mathematical induction for inequalities: $\frac1{n+1} + \frac1{n+2} + \cdots +\frac1{3n+1} > 1$

Prove by induction: $$\frac1{n+1} + \frac1{n+2} + \cdots +\frac1{3n+1} > 1$$ adding $1/(3m+4)$ as the next $m+1$ value proves pretty fruitless. Can I make some simplifications in the inequality ...
5
votes
8answers
474 views

How to show that $\lim \frac{1}{n} \sum_{i=1}^n \frac{1}{i}=0 $? [duplicate]

Show that $$\lim \frac{1}{n} \sum_{i=1}^n \frac{1}{i} =0 $$ I've proved that this sequence converges (it is bounded and decreasing). NOW, I need to find a sequence that is bigger than this one and ...
1
vote
2answers
2k views

Proof by induction of the Inequality of Harmonic numbers: $H_{2^n} \ge 1+ \frac n2$

My question is, for the question below, in the inductive step, where does $\dfrac{1}{2^{(k+1)}}$ come from?And where does $2^k$ come from in the third last step?
1
vote
0answers
61 views

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 ...
43
votes
2answers
1k views

there exist infinite many $n\in\mathbb{N}$ such that $S_n-[S_n]<\frac{1}{n^2}$

recent conjecture :Let $S_n=1+\frac{1}{2}+\frac{1}{3}+\ldots+\frac{1}{n}$, where $n$ is a positive integer. Prove that :there exist infinite many $n\in\mathbb{N^{+}}$ such that $$S_n-[S_n]<\dfrac{1}...
23
votes
4answers
754 views

How find this sum $I_n=\sum_{k=0}^{n}\frac{H_{k+1}H_{n-k+1}}{k+2}$

$$I_n=\sum_{k=0}^{n}\dfrac{H_{k+1}H_{n-k+1}}{k+2}$$ where $$H_{n}=1+\dfrac{1}{2}+\cdots+\dfrac{1}{n}$$ my try:since $$I_n=\dfrac{1+\dfrac{1}{2}+\cdots+\dfrac{1}{n+1}}{2}+\dfrac{\left(1+\dfrac{1}{2}\...
13
votes
2answers
548 views

Sum $\sum^\infty_{n=1}\frac{(-1)^nH_n}{(2n+1)^2}$

I would like to seek your assistance in computing the sum $$\sum^\infty_{n=1}\frac{(-1)^nH_n}{(2n+1)^2}$$ I am stumped by this sum. I have tried summing the residues of $\displaystyle f(z)=\frac{\pi\...
14
votes
6answers
732 views

Proving $\sum_{k=1}^{n}{(-1)^{k+1} {{n}\choose{k}}\frac{1}{k}=H_n}$

I've been trying to prove $$\sum_{k=1}^{n}{(-1)^{k+1} {{n}\choose{k}}\frac{1}{k}=H_n}$$ I've tried perturbation and inversion but still nothing. I've even tried expanding the sum to try and find ...
10
votes
3answers
227 views

Harmonic number identity

I search for an elementary proof of the following identity: $$ \sum_{i=1}^{n-k} \frac{(-1)^{i+1}}{i}\binom{n}{i+k}=\binom{n}{k}\left(H_n-H_k\right) $$ I have found the following proof: $$ \sum_{i=1}^{...
23
votes
1answer
2k views

Formula for the harmonic series $H_n = \sum_{k=1}^n 1/k$ due to Gregorio Fontana.

My question was inspired by this stackexchange question. For the last 90 minutes I have been trying to prove this formula due to Gregorio Fontana: $$H_n = \gamma + \log n + {1 \over 2n} - \sum_{k=2}^\...
9
votes
3answers
538 views

Identity with Harmonic and Catalan numbers

Can anyone help me with this. Prove that $$2\log \left(\sum_{n=0}^{\infty}\binom{2n}{n}\frac{x^n}{n+1}\right)=\sum_{n=1}^{\infty}\binom{2n}{n}\left(H_{2n-1}-H_n\right)\frac{x^n}{n}$$ Where $H_n=\...
5
votes
2answers
130 views

Constant term in Stirling type formula for $\sum^N_{n=1} H_n \cdot \ln(n)$

Is there a similar formula like the Stirling one on the sum over $\ln(n)$ (take logarithms on its factorial representation), $$ \sum_{n=1}^N \ln(n) = N\cdot \ln(N)−N+\frac{\ln(N)}{2}+\frac{\ln(2π)}{2}+...
2
votes
2answers
308 views

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 ...