# Questions tagged [summation-method]

Use for methods for constructing generalized sums of series, generalized limits of sequences, and values of improper integrals.

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### Any ideas on how to evaluate this sum? [closed]

Have you got any ideas on how to evaluate this? $$(x+0)^{20}+(1+x)^{20}+(2+x)^{20}...+(100+x)^{20}$$
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### Summation problem: find k if $\sum_{k=5}^{29} kn-6=1125$ [closed]

How do I find k if $\sum_{k=5}^{29} kn-6=1125$ ? I tried to solve it but couldn’t understand it. Any hints would be appreciated!
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### How do I evaluate this summation series?

I came across this problem and I couldn't solve it $\sum_{r=1}^\infty\frac{6^r}{(3^r-2^r)(3^{r+1}-2^{r+1})}$ So when I saw the solution they wrote $6^r = 3^r(3^{r+1}-2^{r+1}) - 3^{r+1}(3^r-2^r)$ and ...
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### A closed form for $\sum_{k=0}^n \frac{ (-1)^k {n \choose k}^2}{k+1}$

Mathematica gives $$\sum_{k=0}^n \frac{(-1)^k {n \choose k}^2}{k+1}= ~_2F_1[-n,-n;2;-1],$$ where $~_2F_1$ that is Gauss hypergeometric function. Here the question is: Can one find a simpler closed ...
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### Show $\sum_{n \in \mathbb{Z}} \sum_{k \in \mathbb{Z}} a_k b_{n-k} z^n =(\sum_{n \in \mathbb{Z}}a_n z^n)(\sum_{n \in \mathbb{Z}}b_n z^n)$

First, let me tell the definition of series I use. Let $S$ be any set. Let $f: S \to \mathbb{C}$ be a function. We say $\sum_{n \in S}f(n)$ converges to $F\in \mathbb{C}$ if the following condition is ...
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### Add function value over an interval.

To prove: $$\sum_{i=1}^{S}I_i=S \int_{\underline{I}}^{\overline{I}}I f(I)dI$$, where $f(I)$ is a continuous probability density function and $I\in[\underline{I},\overline{I}]$ and $S$ is population . ...
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### Estimate on partial sum $\sum_{n=1}^x \frac{\sin^2(n)}{n}$ using Abel Plana Summation Formula :

I'm trying to get estimate on the following partial summation using Abel-Plana Summation formula: $$\sum_{n=1}^x \frac{\sin^2(n)}{n}$$ I can handle the first integral in the formula but I'm stuck at ...
### On proving that $\sum\limits_{n=1}^\infty \frac{n^{13}}{e^{2\pi n}-1}=\frac 1{24}$
Ramanujan found the following formula: $$\large \sum_{n=1}^\infty \frac{n^{13}}{e^{2\pi n}-1}=\frac 1{24}$$ I let $e^{2\pi n}-1=\left(e^{\pi n}+1\right)\left(e^{\pi n}-1\right)$ to try partial ...
EDIT: Here's a way to remember without thinking hard about it that I came with from the channel of blackpenredpen: $\sum_{k \geq 0} \sum_{i \geq k+1} \mathbb{P}(X=i)$ We have $1\leq \underline{k+1 ... 1answer 116 views ###$ \sum_{n = 1}^{\infty} ( \frac{p_{2n-1}}{p_{2n}} - \frac{p_{2n}}{p_{2n +1}} ) = ?? $Let$ p_n $be the$n$th prime. I was confused about the following idea. $$A = \sum_{n = 1}^{\infty} ( \frac{p_{2n-1}}{p_{2n}} - \frac{p_{2n}}{p_{2n +1}} )$$ Very confused actually. Does this ... 1answer 62 views ### Prove the combinatorial identity For any$n$prove the following identity $$\sum_{i+j+k=n} j \binom{n}{i,j,k} a_{2i+1,2j-1,2k}=\sum_{i+j+k=n} i \binom{n}{i,j,k} a_{2i-1,2j+1,2k},$$ here $$\binom{n}{i,j,k}=\frac{n!}{i! j! k!},$$ is ... 1answer 64 views ### question deleted as question is incomplete. [closed] question deleted as question is incomplete. 1answer 23 views ### what will be the$(t+1)\$-th term in this summation expansion and why? [closed]
What will be the last term in the following summation $$\sum_{k=0}^{t}F_{t}F_{t-1}\cdots F_{k+1}x_k.$$