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.
1,133 questions
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Integrating $\int_0^1 \frac{\ln x \ln (1+x) \ln(1-x)}{x^{\frac12}} \,dx$
$$\int_0^1 \frac{\ln x \ln (1+x) \ln(1-x)}{x^{\frac12}}
\,dx$$
Now I have seen a similar integral here,
I am unable to deal with the $x^{\frac12}$ term in the denominator.
Here's one idea,
$$\int_0^1x^...
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2
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$\sum_{n=2}^{\infty }\big( \sum_{i=1}^n \frac 1i \big) ^2\frac 1{n^2} $ and a related series.
Let $S_1 = \sum_{n=2}^{\infty }\big( \sum_{i=1}^n \frac 1i \big) ^2\frac 1{n^2} $
And $S_2 =\sum_{n=2}^{\infty} \frac {(\ln n) ^{-2P}}{n^P + P^{-(\ln n )^ {\frac 1P} }} $
Consider the following ...
- 2,989
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Properties of a convergent subseries of the Harmonic series?
I have a sequence $(t_k)_{t\in\mathbb{N}}$ in $\mathbb{N}$ such that this subseries of the Harmonic series
$$S_k=\sum_{j=0}^{k-1} \frac{1}{t_j+1}$$
converges, i.e., $\lim_{k\to\infty} S_k < \infty$....
- 119
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1
answer
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Evaluating this intriguing sum of harmonic numbers
Any idea whether a known function can be used to describe this sum:
$$f(n)=\sum_{k=1}^n \frac{H_{k-1}-H_{k-1/2}}{16k^2-1}\,,\tag{1}$$
with $n>0$ and $H_n$ the harmonic numbers? I am interested in a ...
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Evaluating $\sum_{k=1}^{\infty} \frac{(-1)^k \ln(k)}{k}$ [duplicate]
I am trying to evaluate the infinite series
$$
S = \sum_{k=1}^{\infty} \frac{(-1)^k \ln(k)}{k}
$$
Convergence of the sum
I have proven that the series converges using the alternating series test, but ...
- 354
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Proving a summation formula featuring the Harmonic Series and Binomial Coefficients [duplicate]
I want to prove the following:
$$
\sum_{k=1}^n (-1)^{k-1}{n\choose k} \frac{1}{k}=\sum_{k=1}^n\frac{1}{k}
$$
I've tested the n=1, ..., 12 cases and they work, which led me to go for induction but ...
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How to evaluate $-\sum_{n=0}^{\infty} \left( \frac{4^n}{\left( (2n+1) \binom{2n}{n} \right)^2} \right) \frac{H_{n+1}}{n+1}$
Question
How to evaluate
$$
-\sum_{n = 0}^{\infty}
\frac{4^{n}}{\left[\rule{0pt}{5mm}
\left(2n + 1\right)\binom{2n}{n}\right]^{2}}\
\frac{H_{n + 1}}{n +1}
$$
My attempt
Due to another shifted sum
$$
...
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1
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Can sequence 1, 1, 1, 1,.. be called harmonic sequence (beside being geometric and arithmetic)? [closed]
Sequence 1,1,1,1 or for example 5,5,5,5 is arithmetic and geometric, but can it also be harmonical or some other type of sequence?
- 11
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Prove $\sum_{n=1}^{\infty} (2n-1)\left(\frac{\zeta(2)}{2}-\overline{H}_{n-1}^{(2)}\right)^2=\frac{\zeta(2)}{2}$
It is suggested a typical sum involving alternative harmonic-numbers
$$
\sum_{n=1}^{\infty} (2n-1)\left(\frac{\zeta(2)}{2}-\overline{H}_{n-1}^{(2)}\right)^2=\frac{\zeta(2)}{2}
$$
where the alternative ...
- 2,410
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Closed form for a sum with derivatives of the binomial coefficient $\binom{x-m}{n}$
Consider this family of series:
$$\sum _{n=0} ^{\infty} \frac{1}{\left(n+\frac{m}{2}\right)^{\beta}} \frac{d^{\alpha}}{{dx}^{\alpha}} \left. \binom{x-m}{n} \right|_{x=0}, \quad \text{where}~ m \geq0$$
...
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How can I prove both series are equal?
Given $n$ random variables $X_1, X_2,\dots, X_n$, which are independent and exponentially distributed with rate parameter $\lambda$, I was able to prove that
$$\mathbb E[\max\{X_1, X_2,\dots, X_n\}] = ...
- 169
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Summing three binoharmonic series
The following problem
$$\sum_{n=1}^\infty\frac{2^{2n+1}H_n}{n(2n+1)^2\binom{2n}{n}}+\sum_{n=1}^\infty\frac{(H_{n-1})^2}{(2n-1)^2 2^{2n}}\binom{2n}{n}+\sum_{n=1}^\infty\frac{H^{(2)}_{n-1}}{(2n-1)^2 2^{...
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Looking for a general overview of how harmonics on a torus would work
I'm a very early beginner to learning harmonics, and was wondering if someone answer some very general questions I have, since all of the articles I've found online immediately reference advanced ...
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I need help to prove that$\int^1_0 x^{m-1}\ln^2(1-x)dx=\frac{2}{m}\sum_{k=1}^m\frac{H_{k}}{k}$
Show that $$\int^1_0 x^{m-1}\ln^2(1-x)dx=\frac{2}{m}\sum_{k=1}^m\frac{H_{k}}{k}$$
We know:
$$\ln(1-x)=-\sum^{\infty}_{n=1}\frac{x^n}{n}$$
and using cauchy product
Therfore:
$$\begin{align*}
\ln^2(1-x)&...
- 3,260
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Closed form for the partial sums of the Harmonic Series? [closed]
I just realized that the nth partial sum of the Harmonic series is equivalent to the absolute value of the quotient of the coefficient of the linear term divided by the constant term of the polynomial ...
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Is the $d$-dimensional harmonic series is proportional to the surface-area of a $d$-sphere?
TLDR:
How to Prove:
$$ \sum_{\sqrt{a_0^2 + ...+ a_k^2} \le n, (a_0 ,...,a_k) \ne (0)^{k+1} } \frac{1}{(a_0^2 + ...+ a_k^2)^{\frac{k}{2}}} = \frac{k\pi^{\frac{k}{2}}}{\Gamma(\frac{k}{2} + 1)} \ln(n) + ...
- 17.9k
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Is there a closed form for the quadratic Euler Mascheroni Constant?
Short Version:
I am interested in computing (as a closed form) the limit if it does exist:
$$ \lim_{k \rightarrow \infty} \left[\sum_{a^2+b^2 \le k^2; (a,b) \ne 0} \frac{1}{a^2+b^2} - 2\pi\ln(k) \...
- 17.9k
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Does the rate $\log n$ imply "almost harmonic"?
Let $\{c_k\}$ be a decreasing positive sequence such that $\sum_{k=1}^n c_k \sim \log n$. Does it say $c_k=O(1/k)?$
I have found a reference where it says that the converse is true. I tried to tackle ...
- 775
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Showing that $\sum_{i = 1}^N \sum_{j = N - i + 1}^N \frac{1}{ij} = \sum_{i = 1}^N \frac{1}{i^2}$ without induction
It's straightforward to show via induction that
$$
\sum_{i = 1}^N \sum_{j = N - i + 1}^N \frac{1}{ij} = \sum_{i = 1}^N \frac{1}{i^2}\text{.}
$$
(For example, given the table
$$
\begin{array}{|c|c|c|c|}...
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Why Is There No Oscillator Representation for Operators in Planar N=4 SYM Theory?
Im studying the planar N=4 Super Yang-Mills (SYM) theory and I'm curious about the representations of its operators, specifically the Hamiltonian and the dilatation operator. In many quantum systems, ...
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Fractional part of a sum
Define for $n\in\mathbb{N}$ $$S_n=\sum_{r=0}^{n}\binom{n}{r}^2\left(\sum_{k=1}^{n+r}\frac{1}{k^5}\right)$$
I need to find $\{S_n\}$ for $n$ large where $\{x\}$ denotes the fractional part of $x$.
$$...
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Doubt in a question involving summation of a harmonic progression [closed]
I was solving the question involving the sum of harmonic progression
Q) Given that $$\color{red}{g(r) = 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \dots + \frac{1}{r}}$$ and
$$\color{red}{\sum_{r=...
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Application of Bertrand's postulate [duplicate]
We can use Bertrand's Conjecture ( that for any integer $n \not= 0$, there exists at least one prime number $𝑝$ with $n < p \leq 2n$ ) to demonstrate the ...
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Inequality Involving Ratio of Harmonic Numbers
Does there exists a constant $d$ such that for all $n\in \mathbb{N}$ and all $k$-tuples of distinct primes primes $p_1, \ldots, p_k \le n^{1/\log \log n}$, with $k\ge 1$, $$\frac{1}{p_1 \cdots p_k} - \...
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Why does this connection of increasingly exclusive partitions $P_{n,k}$ to the harmonic series $H_k$ exist?
$\mathbf{SETUP}$
In this previous question, I show how the sum of all cycles of type defined by non-unity partitions of $n$ relates to the derangement numbers / subfactorial $!n$ (number of ...
- 107
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answer
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how to evaluate $\sum_{n=1}^\infty \sum_{m=1}^{n-1} \frac{(-t)^{m+n}}{m-n}$?
I tried to find the summation
$$ \Omega=\sum_{n=1}^\infty \sum_{m=1 , m\ne n}^\infty \frac{(-1)^{m+n} H_{m+n}}{m^2-n^2}$$
and got
$$ \Omega=\int_0^1 \frac{\ln(1-t)}{t} \left( \frac{t^2}{1-t^2} \ln(1+t)...
- 1,979
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On properties of the Harmonic numbers and a related coupon collector problem
In a coupon collector problem where we aim to collect $n$ coupons, it takes us in expectation $n H(n)$ trials to collect all coupons, and in expectation, $H(n)$ coupons are collected exactly once (one ...
- 445
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Direct proof that $\sum_{x=1}^n \sum_{y=0}^{x-1} \frac{1}{x(n-y)} = \sum_{x=1}^n \frac{1}{x^2}$
I'm looking for a direct proof that for all $n\in \mathbb N$, $$\sum_{x=1}^n \sum_{y=0}^{x-1} \frac{1}{x(n-y)} = \sum_{x=1}^n \frac{1}{x^2}.$$
It is easy to prove this relation by induction on $n$, ...
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What does the function $f(x) = \lim \limits_{n \to \infty} H_{\lfloor xn \rfloor} - H_n$ look like?
I had a homework question asking me to evaluate the series $1 + \frac{1}{2} - 1 + \frac{1}{3} + \frac{1}{4} - \frac{1}{2}...$
Ultimately the solution was just to combine the negative terms with the ...
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Given constrain $m=a_1>a_2>...>a_n$ and the elements are integer prove $\sum \frac{a_i-a_{i+1}}{a_i} < H_m$
For decreasing positive integers $a_1>a_2>...>a_n>0$ when $a_1=m$.
Mark $a_{n+1}=0$,
Prove that $\sum_{k=1}^n \frac{a_i-a_{i+1}}{a_i} < H_m=\sum_{k=1}^m \frac{1}{k}$
Might add that $n$ ...
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approximation to digamma function
I was learning about the harmonic series back in college to which the professor said "There is no known closed form for the harmonic sum", I felt that was strange given that the sum didn't ...
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vote
1
answer
163
views
How to evaluate $\sum_{n=1}^{\infty}\frac{H_{n}^{(2)}}{n^{4}}$
It may be rather tedious and I will have to delve into deeper, but I have a little something. We probably already know this one. The thing is, the first one results in yet another Euler sum. But, I ...
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How to calculate this sum $\sum_{n=1}^{\infty} \frac{H_n \cdot H_{n+1}}{(n+1)(n+2)}$
How to calculate this sum $$\sum_{n=1}^{\infty} \frac{H_n \cdot H_{n+1}}{(n+1)(n+2)}$$
Attempt
The series telescopes. We have
$$=\frac{H_n \cdot H_{n+1}}{(n+1)(n+2)} = \frac{H_n \cdot H_{n+1}}{n+1} - ...
- 5,957
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vote
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Proving specific properties of $x_{n+1} = x_n + \frac{1}{nx_n}$ where $x_1 = \frac{1}{2}$
First, I am asked to prove that $(x_n)_{n \geq 1}$ is divergent.
Secondly, I have to find the limit of $\left( \frac{x_n}{\sqrt{2\ln(n)}} \right)_{n \geq 1}$ (if it exists).
What I managed to show:
$(...
- 501
4
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2
answers
217
views
Find $\sum^{n}_{k=1} \binom{n}{k} \frac{(-1)^k}{k^a}$
I have been trying to find a general representation of the following finite sum.
$$
S(n,a) = -\sum^{n}_{k=1} \binom{n}{k} \frac{(-1)^k}{k^a}
$$
The sum seems to be related to Generalized Harmonic ...
- 847
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votes
2
answers
166
views
Show that $\sum_{n=1}^{\infty} \frac{\binom{2n}{n} (H_{2n} - H_n)}{4^n (2n - 1)^2} = 2 + \frac{3\pi}{2} \log(2) - 2G - \pi$
Show that $$\sum_{n=1}^{\infty} \frac{\binom{2n}{n} (H_{2n} - H_n)}{4^n (2n - 1)^2} = 2 + \frac{3\pi}{2} \log(2) - 2G - \pi$$
My try :
We know that
$$\sum_{n=1}^{\infty} \binom{2n}{n} (H_{2n} - H_{n}) ...
- 5,957
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1
answer
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views
Ramanujan-Type Harmonic Series $\sum_{r=0}^{\infty}\binom{2r}{r}^3\left(\frac{1/6+r}{2^{8r}}\right)H_{r-1/2}=-\frac{8\ln2}{9\pi}$
We will be considering series such as this :
$$\sum_{r=0}^{\infty}\binom{2r}{r}^3\left(\frac{1/6+r}{2^{8r}}\right)H_{r-1/2}=-\frac{8\ln2}{9\pi}$$
Consider $K(k)$ and $E(k)$ to be the Complete ...
- 3,710
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0
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68
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Why does $\lim\limits_{n\to\infty}\frac{\cosh^{-1}n}{H_n}=1$?
$H_n$ represents the $n\text{th}$ harmonic number. I was messing around with Desmos when I happened to come across this. I typed it into WolframAlpha which confirmed that the limit is equal to $1$ but ...
- 1,930
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votes
2
answers
456
views
Harmonic sum with Dirichlet eta tail
The following problem is proposed by Cornel Valean:
$$\sum_{n=1}^{\infty} \frac{H_n}{2n+1}\left(\eta(2)- \overline{H}_n^{(2)}\right)$$
$$=2 G^2-2\ln(2) \pi G+\ln^2(2)\frac{\pi ^2}{6} +\frac{53}{1440}\...
- 26.6k
1
vote
0
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70
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Approximation of a sum defining the expectation value for Big-O of an algorithm
I am considering an algorithm for sampling from a collection without replacement. The algorithm I am considering is complex in that its runtime explodes under certain conditions. I am not concerned ...
- 11
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votes
2
answers
285
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Asymptotic analysis of $e^{H_n}$
To be precise, what is the asymptotic behavior of $e^{H_n}$, as $n$ tends to infinity, where $e$ is the Euler's number, a mathematical constant approximately equal to $2.71828$ and $H_n$ is the $n$-th ...
- 157
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2
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171
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Closed expression for an integral involving harmonic numbers
Recently, I stumbled on an interesting class of integrals, and here is one example.
Problem: find the closed form of the integral
$$I = \int_{-\frac{\pi }{2}}^{\frac{\pi }{2}} \frac{\cos (x)}{\gamma ^{...
8
votes
1
answer
2k
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Proving $\sum_{n=1}^{\infty}\binom{2n}{n}^2\frac{4H_{2n}-3H_n}{n2^{4n}}=\zeta(2)$
While trying to solve Prove $\int_{0}^{1}\frac1k K(k)\ln\left[\frac{\left(1+k \right)^3}{1-k} \right]\text{d}k=\frac{\pi^3}{4}$
I was able to reduce it to the following form,
$$\sum_{n=1}^{\infty}\...
- 3,710
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votes
0
answers
286
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Solve the integral $\int_0^1 \frac{\ln^2(x+1)-\ln\left(\frac{2x}{x^2+1}\right)\ln x+\ln^2\left(\frac{x}{x+1}\right)}{x^2+1} dx$
I tried to solve this integral and got it, I showed firstly
$$\int_0^1 \frac{\ln^2(x+1)+\ln^2\left(\frac{x}{x+1}\right)}{x^2+1} dx=2\Im\left[\text{Li}_3(1+i) \right] $$
and for other integral
$$\int_0^...
- 1,979
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2
answers
135
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The Expected Value of "Descending Dice Problem" and Harmonic Numbers
I named the "Descending Dice Problem", but not sure if there is another name to it. This is how it goes:
You start with a N sided dice. Throw it, suppose you rolled R as your result, and ...
- 73
4
votes
1
answer
157
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Simpler proof of identity $ \sum_{n=1}^{\infty} \frac{H_n^2}{n2^n} = \frac{7}{8} \zeta(3) $
The identity in question can be obtained by first proving
$$ \sum_{n=1}^{\infty} \frac{H_n^2}{n} z^n = - \frac{1}{3} \log^3(1-z) -\log(1-z) \text{Li}_2(z) + \text{Li}_3(z), \hspace{0.5cm} |z|<1, $$
...
- 1,773
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0
answers
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Wolfram wishlist: series of $\Gamma(z)$ in $z=-m$. Cycle index of symmetric groups.
I found out that Wolfram has a wish list of formulas it is researching.
The first point is "Series for the gamma function"
We are searching for general formulas for the series expansion of ...
4
votes
1
answer
131
views
How to compute the sum $\sum_{n\geq 1}(-1)^n\Big(H_n^2-(\gamma+\log(n))^2\Big)$
How to compute the sum $$\sum_{n\geq 1}(-1)^n\Big(H_n^2-(\gamma+\log(n))^2\Big).$$
I had computed some similar sums
$$\sum_{n\geq 1}\frac{H_n-\log(n)-\gamma}{n},\qquad
\sum_{n\geq 1}(-1)^n (H_n-\log(n)...
- 1,271
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1
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Help me ask this confused question about the divergence of the harmonic series
I've struggled for a couple of weeks to figure out the question I really want to ask, but without any success, so I'm just putting out this rather confused question out in hopes that someone can help ...
- 66.8k
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4
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Harmonic series multiplied by $-2$ every third term converges to $\ln 3$
Prove that the series
$$\sum_{n=1}^\infty a_n = 1+\frac 1 2 -2\cdot \frac 1 3+\frac 1 4 + \frac 1 5 - 2\cdot \frac 1 6 + \cdots $$ converges to $\ln (3)$.
Inserting parentheses every three terms (...
- 750