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.

582 questions
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Finding $f(x)$ that makes the limit $\lim_{x \to \infty} g(x) = e$ and Satisfies Other Condtions

So, for this proof I am working on, I encountered this equation: $$g(x)=(e^{H(x)} \cdot H(x)^{({e^{H(x)}})})^{f(x)}$$ where $H(x)$ is the harmonic series-- $\sum_{n=1}^{x}\dfrac{1}{n}$. So, for me to ...
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Prove $\frac{\partial}{\partial m}\text{B}(n,m)=-\text{B}(n,m)\sum_{k=0}^{n-1}\frac{1}{k+m}$

where $\ \displaystyle\text{B}(n,m)=\int_0^1 x^{n-1}(1-x)^{m-1}\ dx=\frac{\Gamma(n)\Gamma(m)}{\Gamma(n+m)}\$is the beta function, defined over positive $\ n,m>0$. The point of this post is to ...
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Finding $f(x)$ that makes the limit $\lim_{x \to \infty} g(x) = e$ [on hold]

So, with this proof I am working on, I have this equation: $$g(x)=(e^{H(x)}*H(x)^{({e^{H(x)}})})^{f(x)}$$ where $H(x)$ is the harmonic series-- $\sum_{n=1}^{x}\dfrac{1}{n}$. So, for me to continue, I ...
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A “binomial” generalization of harmonic numbers

For positive integers $s$ and $n$ (let's limit the generality), define $$H_s(n)=\sum_{k=1}^{n}\frac{1}{k^s},\qquad G_s(n)=\sum_{k=1}^{n}\binom{n}{k}\frac{(-1)^{k-1}}{k^s}.$$ The former is well-known; ...
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evaluate $\sum_{n=1}^\infty\frac{(-1)^{n-1}H_n}{(2n+1)^3}$

where $H_n=1+\frac1{2}+\frac1{3}+...+\frac1{n}$ is the $n$th harmonic number. this sum was proposed by Cornel and I solved it using integration. can be solved using series manipulation? here is the ...
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Is there a closed form for a give infinite sum?

I've been asked to evaluate this sum $$\sum_{n=0}^{\infty}\frac{C_n^2}{2^{4n}}(H_{n+1}-H_n)$$ where $C_n=\frac1{n+1}{2n\choose n}$ denotes the $n$th Catalan number and $H_n$ denotes the nth Harmonic ...
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How to show $\sum_{n=1}^{\infty}\frac{H_{n+1}}{n(n+1)}=2$?

I am interested in the proof of $$\sum_{n=1}^{\infty}\frac{H_{n+1}}{n(n+1)}=2, \quad H_{n}=1+\frac{1}{2}+\cdots+\frac{1}{n}$$ This result can be verified by Mathematica or by WolframAlpha This ...
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Calculating the summation$\sum_{n=1}^{\infty}\frac{H_{n+1}}{n(n+1)}$

I need to find explicitly the following summation $$\sum_{n=1}^{\infty}\frac{H_{n+1}}{n(n+1)}, \quad H_{n}=1+\frac{1}{2}+\cdots+\frac{1}{n}$$ From Mathematica, I checked that the answer is $2$. The ...
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Okay, so the limiting difference between the harmonic series and the natural logarithm is known as the Euler-Mascheroni constant, $\gamma= 0,577$. My question is: is there any base for the logarithm ...
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Find limit of sequence defined by sum of previous terms and harmonics

I came across this sequence as part of my work. Could someone indicate me the methodology I should follow to solve it? I guess it involves harmonic numbers and/or the digamma function? I tried to ...
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A series involves harmonic number

How do we get a closed form for $$\sum_{n=1}^\infty\frac{H_n}{(2n+1)^2}$$
Compute $\sum_{k≥1,n≥1}\frac{(-1)^{n+k}}{k(k+1+n)^2}$
How can Compute in closed form this double summation : $$\sum_{k≥1,n≥1}\frac{(-1)^{n+k}}{k(k+1+n)^2}$$ I think here can use harmonic series Actually I don't have any ideas to approach it