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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).

Questions about the Euler-Maclaurin summation formula. For questions about Euler's formula, consider using the tag instead. For questions about Maclaurin series, use .

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139 questions
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Exact result of a series using Euler-Maclaurin expansion.

This is a variant of Exercise 64 in Chapter 9 of concrete mathematics. Prove the following identity \begin{equation} \sum_{n = -\infty}^{\infty}' \frac{1 - \cos( 2\pi n k )}{n^2 } = 2 \pi^2 ( k - k^2 ) \qquad k \in [0,1] \end{equation} I came…
anecdote
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2 answers

Euler-Maclaurin summation for $e^{-x^2}$

I want to approximate the sum $$\sum_{k=0}^\infty e^{-k^2}$$ using the Euler-Maclaurin formula $$\sum_{k=0}^\infty f(k) = \int_0^\infty f(x) \, dx + \frac{1}{2}(f(0) + f(\infty)) + \frac{1}{12}(f'(\infty) - f'(0)) - \frac{1}{720}(f'''(\infty) -…
user136700
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Using the Euler-Maclaurin formula to approximate Euler's constant, $\gamma := \lim_{n\to\infty}\left(-\ln n+\sum_{k=0}^n\frac1n\right)$

Let $\gamma=\lim_{n\to\infty} F(n)$ where $$F(n)=1+\frac{1}{2}+\frac{1}{3}+\cdots\frac{1}{n}-\ln(n)$$ (This is Euler's constant.) How can I calculate $\gamma$ with $10$ digits of precision using the Euler-Maclaurin Formula?
mathie12
  • 211
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How can $i^i = e^{-\pi/2}$ !!

I was asked a homework question: find $i^i$. The solution provided was as follows: Let $A = i^i$. $\log A = i \log i$. Now, $\log i = \log e^{i\pi/2} = \frac{i\pi}{2}$. So, $\log A = -\frac{\pi}{2}$ Thus, $i^i = e^{-\pi/2}$. I understood how the…
Truth-seek
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Finding an asymptotic expansion for $\sum_{k=0}^{n} \frac{1}{1+\frac{k}{n}}$

It is well known that an asymptotic expansion of the n-th harmonic number is $$H_{n}= \sum_{k=1}^{n} \frac{1}{k} \sim \ln(n) + \gamma + \frac{1}{2n} -\frac{1}{12n^{2}} + O(n^{-4}).$$ How could we find an asymptotic expansion for the sum $…
cinvro
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Solving a system of differential-like equations for Reverse Euler-Maclaurin Summation

Aim A particular instance of a rational zeries that has as of yet not been evaluated is: \begin{align} Z:= \sum_{n=1}^{\infty} \frac{\zeta(2n)}{(2n)!}. \label{EM1} \tag{EM1} \end{align} This sum can be found in equation (141) of this page.…
Max Lonysa Muller
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The asymptotic $\prod_{k=1}^n\int_{(k-1)/n}^{k/n}f(t)\mathrm{d}t\sim A/(Bn)^n$

Let $P_f(n)=\prod_{k=1}^n\int_{(k-1)/n}^{k/n}f(t)\mathrm{d}t$ be the product of integral parts of a function $f$. For simplicity, assume $f$ is smooth and positive-valued on $(0,1)$. I noticed some limits in this question and this question can be…
coiso
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Do we have to use Bernoulli polynomials in the Euler-Maclaurin summation formula?

It may be that I have not picked up the proof, but I cannot see where the third condition of Bernoulli polynomials, given below, is used in the derivation of the Euler-Maclaurin summation formula. The Bernoulli polynomials are defined inductively…
Sputnik
  • 3,887
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2 answers

Compute $\sum \frac{1}{k^2}$ using Euler-Maclaurin formula

I read that Euler used the summation formula to calculate the value of the series $\sum_{k =1}^{\infty} \frac{1}{k^2}$ to high precision without too much hassle. The article Dances between continuous and discrete: Euler’s summation formula goes into…
Timm von Puttkamer
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A derivation of the Euler-Maclaurin formula?

The generating function for the Bernoulli numbers $B_n$ is $$\frac{x}{e^x-1}=\sum_{n=0}^\infty\frac{B_n}{n!}x^n$$ The sum of an infinite geometric series is $$\frac{1}{1-x}=\sum_{k=0}^\infty x^k$$ Replacing $x$ with $e^x$…
user76284
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1 answer

eulers original derivation for the Euler–Maclaurin formula?

Please does someone know a good description of how Euler did derive his summation formula? Thank you!
thatseries
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1 answer

Asymptotic expansion when E-M formula fails

I want to compute the asymptotic expansion of $$ \sum_{k=1}^n\sqrt{1+t\sin(2\pi k/n)} $$ when $n\to\infty$ and parameter $0
Covariant
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Formula for $f(1) + f(2) + \cdots + f(n)$: Euler-Maclaurin summation formula

Let $f\colon \mathbb{R}\to \mathbb{R}$ be a function with $k$ continuous derivatives. We want to find an expression for $$ S=f(1)+f(2)+f(3)+\ldots+f(n). $$ I'm currently reading Analysis by Its History by Hairer and Wanner. They first consider the…
Sebastian
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Euler-Maclaurin Summation

Using EM summation formula estimate $$ \sum_{k=1}^n \sqrt k $$ up to the term involving $\frac{1}{\sqrt n}$ My attempt is $$ \sum_{k=1}^n \sqrt k = \frac{2 \sqrt{n^3}}{3} -\frac{2}{3} + \frac 1 2 (\sqrt n -1)+ \frac{1}{24} (\frac{1}{\sqrt n} -1)…
Bryce Ramgovind
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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 $S \approx 0.226354\ldots$. Since the terms decrease monotonely in absolute…
Yuriy S
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