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.
1,178 questions
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What is the 'easiest' way to prove the sum of squared reciprocals converges? [duplicate]
By 'easy' I mean avoiding heavy notation. Ideally we only need numbers and multiplication signs. You can use algebra implicitly, but try to keep things elementary school level.
Two examples of what I ...
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2
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Continued fraction with harmonic numbers [closed]
Let $H_n$ be the $n$th harmonic number defined by:
$$ H_{n} := \sum_{k=1}^{n} \frac{1}{k} .$$
Moreover, define $S$ as the limit of the following continued fraction:
$$ S := \cfrac{1}{H_{1}+ \cfrac{1}{...
- 8,064
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Direct proof of the identity $\sum_{j=0}^{s}\binom{s}{j}^2H_j^{(2)}=\binom{2s}{s}\left(2H_s^{(2)}-3\sum_{k=1}^{s}\dfrac{1}{k^2\binom{2k}{k}}\right)$?
Notation:
Pochhammer symbol $(x)_{n}=\dfrac{\Gamma(x+n)}{\Gamma(x)}$,
Generalized Harmonic numbers $H_n^{(r)}\displaystyle=\sum_{k=1}^{n}\dfrac{1}{k^r}$.
Context:
I tried to decompose $\displaystyle\...
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A possible exponential generating function for the generalized harmonic numbers $H_n^{(p)}$
Short primer on two functions which motivate my question. The generalized harmonic numbers are defined as
\begin{align}
H_n^{(p)}:=\sum_{k=1}^n\frac{1}{k^p}.
\end{align}
They admit the (ordinary) ...
- 2,873
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77
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Generalized Partial Harmonic Sum
I am trying to approximate multi-dimensional partial harmonic series $H_n(t)$ to hopefully arrive at a generalized formula similar to the Euler-Maclaurin formula for the 1-dimensional case. Any advice ...
- 157
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Asymptotic Behavior of an Alternating Modified Harmonic Sum Series
Motivation: As a personal side project I have been working with an inclusion-exclusion formulation that is counting weighted power’s $x^a$ between consecutive squares $[n^2, (n+1)^2]$. The function $f(...
- 157
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Finding the closed form for sequences related to Harmonic numbers: -1, 3, -37/9, 49/9...
I encountered three sequences.
The first sequence is explicitly defined using Harmonic numbers $H_m$:
$$\text{Seq 1: } \quad \frac{1}{8}, -\frac{3}{16}, \frac{11}{48}, -\frac{25}{96}, \dots$$
The ...
- 1
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votes
2
answers
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Solving a differential equation of infinite order
I would like to solve the following differential equation
$$\frac{1}{x+1} = \sum_{n=1}^{\infty} \frac{f^{(n)}(x)}{n!}.$$
Does anyone know how to solve this, or if a solution even exists?
My guess is ...
- 111
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Infinite Sums Containing Alternating Euler Sum for Odd Powers [duplicate]
When I was evaluating this monstrous integral $$
\int_0^{\frac{\pi}{2}} x^3 \ln^2 \left(\sin x\right) \, \mathrm{d}x
$$
I managed to reduce it using the fact that $$
\ln^2\left(\sin x\right) = \frac{\...
- 175
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Reference request for some sums
I have been looking at sums with binomial coefficients in their denominator. These are extensions of Apery's series, which he used in his proof of the irrationality of $\zeta(3)$. This weekend I ...
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2
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Find $\sum\limits_{i,j,n\ge1}\frac{n + j + i}{n j i (n + j)(n + i)(j + i)}$
how to find the following series:
$$\sum_{i,j,n\ge1}\frac{n + j + i}{n j i (n + j)(n + i)(j + i)}$$
what i attempted was using symmetry like this
\begin{align*}
\sum_{i,j,n \ge 1} \frac{n + j + i}{n j ...
- 93
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Inequality involving products with harmonic numbers
Let
$$
H_n = \sum_{k=1}^n \frac{1}{k}, \qquad n \ge 1,
$$
and for a fixed parameter $r \in (0,1]$, define a sequence $(a_j)_{j\ge 1}$ by
$$
a_j = 1 - \frac{r}{j}\bigl(H_{j+1}-1\bigr), \qquad j \ge 1.
$...
- 1
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How to evaluate $\displaystyle\sum_{n=1}^\infty\frac{\overline{H}_n}{(2n+1)^6}$ [closed]
How can you evaluate $\displaystyle\sum_{n=1}^\infty\frac{\overline{H}_n}{(2n+1)^6}=\sum_{n=1}^\infty\frac{\overline{H}_n}{n^6}-\frac{1}{64}\sum_{n=1}^\infty\frac{\overline{H}_{2n}}{n^6}$,
where $\...
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votes
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Optimal way to stack blocks for maximum overhang
The figure shown above shows the optimal way for stacking 30 blocks to get the maximum overhang. How does one verify/prove that this shape is indeed the best way to stack the blocks to achieve the ...
- 135
1
vote
1
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Does Taylor series exist for this function?
I recently tried to approximate Harmonic numbers $H(n) $ for very large values of $n$ and that is when I accidentally came across this.
My method of approach was to express $H(n)$ as a continuous ...
- 154