# 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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### Prove using taylor's thorem . x − (x ^3 )/3! ≤ sin x ≤ x − (x ^3)/ 3! +( x^ 5)/ 5! , x ≥ 0. [closed]

Prove using taylor's theorem. x − (x ^3 )/3! ≤ sin x ≤ x − (x ^3)/ 3! +( x^ 5)/ 5! , x ≥ 0.
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### Proving Euler-Maclaurin approximation for even functions using Poisson summation

I'd like to ask for help for the following from my Advanced Mathematics for Physics class (6th semester): If $f(x)$ is a sufficiently regular and even function integrable in all $\mathbb{R}$, use ...
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### Find the Maclaurin Series for this particular function

Let a value $x_1 \in \mathbb{R}$, $x_1 > 0$ such that $\sin(x_1)=\sin(x_1^2)$. Next, \begin{equation*} f(x) = \left\{ \begin{array}{ll} -\sin(x) & x \leq -x_1 \\ ...
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### $n$-th degree Maclaurin vs $n$-th Maclaurin poly

What is the difference between $n$-th degree Maclaurin vs $n$-th Maclaurin poly? Say I had the Maclaurin for $e^{-x^2}$. To find the fourth-degree polynomial, would I go up to $x^4/2$ or continue onto ...
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### Find the development of a function $f(x)= x\cos^2(2x)$ in the power series in point $0$, and after that find derivative of $f^{(21)}(0)$

I have to find the development of a function $f(x)= x\cos^2(2x)$ in the power series in point $0,$ and after that find derivative of $f^{(21)}(0)$ I have started with:\begin{align}f(x)&=\frac{x(\...
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### Difference between sum and integral of $\frac{1}{(x^2 + a^2)^2}$

I am trying to use the Euler-Maclaurin formula to obtain the following difference: \begin{align} \Delta &= \sum_{n=1}^\infty f(n) - \int_0^\infty dx f(x) \\ &= \frac{ f(\infty) - f(0) }{2} + ...
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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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### 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 ...