# 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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### How to express $\sum_{i=0}^{m} \exp [(\frac{a}{b+c+i})^2]$ in terms of an integral?

I have this sum $$\sum_{i=0}^{m} \exp [(\frac{a}{b+c+i})^2]$$ where the upper limit $m$ is a finite non-negative integer, and $a,b,c\in\mathbb{R}$. I want to transform summation to an integral ...
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### Exercise I.0.9 from Tenenbaum's "Introduction to analytic and probabilistic number theory"

I'm trying to solve all the exercises from Tenenbaum's book but am unfortunately stuck on problem 9 of the very first ("tools") chapter. The problem is supposed to be an application of the ...
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### Finding the Maclaurin series for the given function.

I am dealing with a problem that says:Find the Maclaurin series for the function $f(x)=\ln(1+x+x^2)$. I have tried to write the expression inside the brackets exactly, $1+x+x^2$ as a product of two ...
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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 ...
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### Asymptotic approximations for $\Phi(z,-n,0)_\nu=\sum_{k=0}^{\nu-1}k^nz^k$ as $\nu\to\infty$

I am interested in asymptotic approximations of $$\Phi(z,-n,0)_\nu:=\sum_{k=0}^{\nu-1}k^nz^k=\Phi(z,-n,0)-z^\nu\Phi(z,-n,\nu),\quad n\in\Bbb N$$ for large $\nu$ where $\Phi(z,s,a)$ is the Lerch ...
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### Euler–Maclaurin formula: Mismatched dimensions

To quote Wikipedia: If ${\displaystyle m}$ and ${\displaystyle n}$ are natural numbers and ${\displaystyle f(x)}$ is a real or complex valued continuous function for real numbers ${\displaystyle x}$ ...
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### Substitution variables in Taylor series

I have troubles understanding why and when you can substitute your variables in a Taylor series. Could somebody help me explain why that is possible? Especially because the derivative often involves ...
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### Finding Maclaurin series f(x)

Can anyone please help me with finding Maclaurin series for this $$f(x) = x^3 \tan^{-1}(2x); \quad |x|<\frac12$$ https://i.sstatic.net/bUhxk.jpg
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### Maclaurin series approximate problem

Use the Maclaurin series to find an approximate value for the following integral: $$\int_{1}^{12} \sin(x^{4})dx$$ Need help with this question please
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### MCQ: comparing two quantities; $\tan(1)$ and $\frac{\pi}{2}$ [duplicate]

Without using a calculator, given that $\pi \approx 3.14159$, compare: FIRST VALUE: $\tan(1)$ [the angle is in radians]. SECOND VALUE: $\frac{\pi}{2}$ Options: (A) FIRST VALUE is greater than the ...
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### Infinite sum of cosines as a Gaussian

I have the following function $$P(x) = \frac{1}{L} + \frac{2}{L} \sum_{n = 1}^{\infty} \cos\left( \frac{n \pi x}{L} \right)$$ and I try to show that it represents a ...
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### Taylor & Maclaurin polynomials number of times you should derivate

I'm trying to understand how many times you should derivate the function which you are working with when using Maclaurin or Taylor. To my understanding the more times you do it the more accurate it ...
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### evaluation of $\sum_{n=-\infty}^{\infty}e^{-2\pi^{2}z^{2}n^{2}}$ for large z

I want to evaluate the following $$\sum_{n=-\infty}^{\infty}e^{-2\pi^{2}z^{2}n^{2}}$$ I know for $z\ll 1$ we can use Euler-Maclaurin formula but in my case z is quite ...
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