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5 questions
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Questions about iterating the Euler-Maclaurin summation formula
Introduction
The Euler–Maclaurin summation formula is as follows for a positive integer $p$ and a continuous function $f(\cdot)$ that is $p$ times continuously differentiable on the interval $[m,n]$ : ...
2
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Are there any published studies on cases of infinite sums for which the Euler–Maclaurin summation method yields the exact evaluation?
The Euler–Maclaurin summation formula is as follows: $$\sum_{i=m}^{n} f(i) = \int_{m}^{n} f(x) dx + \frac{f(n)+f(m)}{2} + \sum_{k=1}^{\lfloor p/2 \rfloor} \frac{B_{2k}}{(2k)!}\big{(}f^{(2k-1)}(n)-f^{(...
4
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2
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From Zurab's integral representation for the Apéry's constant to almost impossible integrals
I would like to know if the following integrals are known, or in case that aren't in the literature we can calculate these in closed-form (in terms of elementary and standard functions). I wondered ...
5
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Two integral representations for $\zeta(3)$ from Zurab's integral and standard formulas for the gamma function
This morning I wrote with the help of a CAS, and integral representation for the Apéry's constant $\zeta(3)$ and some standard formulas two formulas involving this constant. I would like to know if ...
5
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373
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sum, integral of certain functions
While working on some research, I have encountered an infinite series and its improper integral analogue:
\begin{align}\sum_{m=1}^{\infty}\frac1{\sqrt{m(m+1)(m+2)+\sqrt{m^3(m+2)^3}}}&=\frac12+\...