# 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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### write a closed form of $\sum_{1=k}^n i/2^i$

Ive been thinking about this problem for a while now and i cannot understand if there is an actual typo in the equation or its supposed to be like this. I havent been given any value for i. Thats why ...
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### Polygamma sum problem ...

Hello guys i have a problem evaluating the following sum $$\sum_{n=1}^{+\infty}\frac{n(n+1)}{2}\frac{4x(3\pi ^{2}(n+1)^{2}+x^{2})}{(x^{2}-\pi ^{2}(n+1)^{2})^{3}}$$ It is obviously of the polygamma ...
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### Polygamma sum problem

I have a problem evaluating the following sum, $$\sum_{n=1}^{+\infty}\frac{4nx(3\pi ^{2}(n+1)^{2}+x^{2})}{(x^{2}-\pi ^{2}(n+1)^{2})^{3}}$$ The sum obviously is of the form of a polygamma function. ...
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### Doubt regarding Abel Plana summation formula

I have a doubt regarding the Abel Plana summation formula. What is the condition to apply the Abel Plana summation formula.?? This question arises because sometimes while evaluating series it gives ...
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### Does my odd proof for the Abel sum for $\eta(-2)$ work?

EDIT: The correct answer to the Abel sum of $\eta(-2)$ has been given by the comments under this post. The focus of the question is now whether there is any sense to my method and my "proof" ...
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### Prove that the infinite sum of the difference of even and odd values of the Riemann zeta function is 1/2

I am interested in finding closed form solutions for the positive odd integers of the Riemann zeta function, of which only 1 is known. Please forgive me if this is already proven or well-known, but in ...
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### What is $\sum_{3\leq p\leq x} \pi(\sqrt{p})$?

What is the $\displaystyle \sum_{3\leq p\leq x} \pi(\sqrt{p})$? I thought about starting from $\displaystyle 2\sum_{3\leq p\leq x}\frac{\sqrt{p}}{\log p}$.
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### How to go from this summation to this approximation?

https://learn.fmi.uni-sofia.bg/pluginfile.php/194197/mod_resource/content/2/Telephone_numbers.pdf I'm a high school student investigating on telephone numbers (involution numbers) depicted as T(n). If ...
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### Are there any good examples from other math fields or intuition supporting $\int_0^1\frac1xdx=\int_1^\infty\frac1xdx$?

This question is related to the potential possibilities of classification of divergent integrals more precisely than just "divergent to infinity" and the like. Improper divergent integrals ...
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### Euler's summability method for series convergence

I was going through the book on Functional Analysis by Erwin Kreyzig, and I came across this as one of the exercises. As an application of functionals to summability of sequences, the following ...
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I do not understand how to get asymptotics for the double sum $$\sum_{ab^2 < x} ab^2$$ If I sum over $a$ first, I get $$\sum_{b^2<x} b^2 \sum_{a < x/b^2} a = \frac12 \sum_{b^2<x} b^2 \frac{... 1answer 102 views ### Double summation index notation: \Sigma_{i<j} versus \Sigma_{i\neq j}? What is the difference between the summations using i<j and i\neq j in the formula below:$$\sigma^{2}(\boldsymbol{w})=\sum_{i} \tilde{w}_{i}^{2}+2 \sum_{i<j} \tilde{w}_{i} \tilde{w}_{j} \...
If the summation just sum every term i was thinking that for instance 1+2+3+4 = 4+3+2+1 so why this $$\sum\limits_{i=1}^{n} (3i)\ = \sum\limits_{i=n}^{1} (3i)$$ is not true ? And how i can ...