# Questions tagged [euler-maclaurin]

Questions about the Euler-Maclaurin summation formula. For questions about Euler's formula, consider using the tag (complex-numbers) instead. For questions about Maclaurin series, use (taylor-expansion).

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### What is the relation between restoring force and maclaurin series?

I have seen in some of the books any kind of restoring force can be expressed in maclaurins series about the equilibrium position. And they always seem to pick the second term as the first one is ...
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### High precision evaluation of the series $\sum_{n=3}^\infty (-1)^n (1-n^{1/n})$

This series converges conditionally, but it's quite slow. I would like to find its value with high accuracy: $$S=\sum_{n=3}^\infty (-1)^n (1-n^{1/n})$$ Wolfram Alpha gives $S \approx 0.226354\ldots$....
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### Asymptotic formula for $\sum_{k=N}^\infty \frac{x^k}{k!}$ as $N \to \infty$

This seems like a weird question, because the series has good convergence and there's no need to use other methods to estimate it for $N \to \infty$. However, after seeing this question, I tried to ...
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### Maclaurin series expansion of $f(x) = \ln(3x^2 +4x +1)$ [closed]

Can someone please explain how I do the following Maclaurin series? $$f(x) = \ln(3x^2 +4x +1)$$
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### How do I use the Euler-Maclaurin Summation to find the zeros of the Riemann-Zeta function?

I read this paper which describes Euler-Maclaurin Summation as a method for evaluating sums in terms of an integral. I also read from this Quora answer that Euler-Maclaurin summation is one method to ...
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### Property of the remainder term in the Euler-Maclaurin formula for $\sum_{i=1}^n\log i$.

From https://en.wikipedia.org/wiki/Stirling's_approximation, I'm having trouble understanding $$R_{m,n}=\lim_{n\to\infty} R_{m,n}+O(n^{-m}).$$ I worked out $$R_{m,n}=\int_1^{n}\frac{P_m(x)}{mx^m}~ dx$$...
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### I want to learn about -1/12 [duplicate]

I am completely fascinated by the Euler/Ramanujan result $$\sum_{n=0}^{\infty} n = -\frac{1}{12}$$ It is amazing to me that there are so many seemingly bogus ways to evaluate this, and they all ...
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### Euler-Maclaurin Formula Definition Confusion

I am confused about these $2$ definitions of the Euler-Maclaurin formula. I read the following here: The full Euler-Maclaurin formula with no remainder term (for infinitely differentiable $f$ ) is ...
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### Solving Maclaurin series

I have to tackle the following question. My thoughts so far are below it. (A) By successive differentiation find the first four non-zero terms in the Maclaurin series for $$F(x)=(x+1)\ln(1+x)-x$$ (...
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### The nth term of the maclaurin sequence of $\frac 1{1+x+x^2}$

$$\frac 1{1+x+x^2}$$ $$= \sum^\infty_{n=0} {(-1)}^n{(x+x^2)}^n$$ $$= \sum^\infty_{n=0}{(-x)}^n \sum^n_{k=0} {_nC_k}x^k$$ $$= \sum^\infty_{n=0}\sum^{[\frac n2]}_{k=0}{(-1)}^{n-k}{_{n-k}C_k}x^n$$ I ...
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### Proof That $\sum_{k=1}^{n-1}\int_{k}^{k+1}\left\{x\right\}f´(x)dx=\int_{1}^{n}\left\{x\right\}f´(x)dx$

I am Reading the following notes: Ramanujan summation of divergent series by B Candelpergher (https://hal.univ-cotedazur.fr/hal-01150208v2/document). There, the author derives the Euler-MacLaurin ...
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### Anyone knows how to calculate the sum of this series?

$$\sum_{n=1}^{999} \log_{10}\left(\frac{n+1}{n}\right)$$ Can anybody help me how to calculate this summation?
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### Integral involving the fractional part function

Let $\{\}$ denote the fractional part function and $s>1$ be a real number, then does the following integral admit a closed-form ? $$\int_{0}^{1}\bigg\{\frac{1}{x^s}\bigg\}dx\,$$
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### How to integrate the “fractional part”/sawtooth function?

Let us define $\{t\} = t - \lfloor t \rfloor$ this is also sometimes referred to as frac$(t)$. With this in mind how would I calculate $$I := \int_a^b f(\{t\}) dt$$ for some function $f$? I ask ...
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### 6th Degree Polynomial and Chebyshev minmax

I have to find the 6th degree polynomial for the function $f(x)=xe^x$. After which the use of the Chebyshev min-max approach I have to use the list grade polynomial approach with respect to the fault ...
I need to find $\arctan(3)$ and before this, I'm asked to find the Maclaurin series for $\arctan(x)$, so I know they must be related. But $\arctan(x)$ is defined with $x \in [-1,1]$ so what can I do?