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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).

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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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Question on transforming a sum to an integral using the Euler–Maclaurin formula.

I have a question regarding transforming a summation to an integral using the Euler–Maclaurin formula. Imagine I have this sum $$\sum_{i=0}^{m} f(i) \qquad \text{with} \qquad f(i)= \exp [(\frac{a}{b+...
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Does the approximation related to the generalized quadratic Gauss sums hold?

I have a generalized quadratic Gauss sums, which is defiend as $$S(a,b) = \sum_{n = 0}^Ne^{-j(an + bn^2)}$$ where $a\in(-\pi,\pi)$, $b\in(0,\pi)$ and $n$ is an integer. Now I am trying to approximate ...
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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 ...
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Why series expansion doesn’t work

The Euler Maclaurin Formula is $$\sum_{k=0}^x f(k)=\int_0^xf(t)dt+\frac{f(x)+f(0)}2+R_1\tag{1}$$Where $R_1$ is the remainder term defined as (according to Wikipedia)$$R_p=\frac{(-1)^{p+1}}{p!}\int_0^...
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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 ...
confused's user avatar
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What functions satisfy Euler-Maclaurin Summation formula and approach zero?

Wikipedia states that the Euler-Maclaurin formula takes two versions. Here is the one that I am interested in: $$\sum_{k=m}^nf(k)=\int_m^nf(x)dx+\frac{f(n)+f(m)}2+\sum_{k=1}^{\lfloor\frac{p}2\rfloor}\...
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Maclaurin series without Bernoulli or Euler numbers?

I've been working through these listed Maclaurin series and deriving all of them. For the listed trigonometric functions, I'm wondering if there are ways to do things: not have Bernoulli or Euler ...
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New series expansion formula?

Let $f(x)$ be a continuous function that converges to $0$ and can be differentiated for an infinite amount of times. Then we have $$f(x)=\sum_{k=1}^\infty\left(\frac{(-1)^kB_{2k}}{(2k)!}f^{(k+1)}(x)-f'...
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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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Euler - Mc Laurin summation degree of precision

I'm try to know if the precison of Euler-McLaurin summation depends by the index N inside the formula as follow: $\zeta(s)_N = \sum_{k=1}^{N} k^{-s} + \frac{N^{1-s}}{1-s} + \int_{N}^{\infty} \frac{x-[...
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Is it always true that $\sum\limits_{j=1}^n\frac{1}{j} > \int_1^{n+1} \frac{dt}t$?

This inequality was presented as obvious in this answer. I was interested in understanding the details for this comparison so I started reading up the Euler-MacLaurin formula but this actually ...
Larry Freeman's user avatar
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Is there a way to find the square of an arctan MacLaurin series without manually calculating the derivatives?

I got this question from my calculus 2 textbook: We are also given this known series: Is there anyway to calculate arctan^2(x) using this table of already-solved ...
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Euler Maclaurin of a function with discontinuous derivative .

I need to find the Euler Maclaurin summation of a function that can be generalized as the sinc function. $$\frac{\sin(f(x))}{f(x)}$$ But the sinc's function derivative is undefined at zero, which is ...
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Is there a way to compute an asymptotic formula of $\sum_{n\leq x}\lceil 2\sqrt{n} \rceil$? [closed]

I was trying to find an asymptotic formula for the sum $$ \sum_{n\leq x}\lceil 2\sqrt{n} \rceil$$ using Euler–Maclaurin summation formula.. but I really don't know how to proceed in this case, mainly ...
MathRevenge's user avatar
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What does the author want to say? ("An Elementary View of Euler's Summation Formula" by Tom M. Apostol)

I am reading "An Elementary View of Euler's Summation Formula" by Tom M. Apostol. Let $f$ be a function with a countinuous derivative on the interval $[1,n]$. We consider $\sum_{k=1}^{n-1}f(...
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Euler Maclaurin formula for functions with singularities

Is there a version of the Euler Maclaurin formula for treating functions having singularities ? for example $$ \sum_{k=0}^{99} \sum_{x=0}^{n} \frac{1}{\frac{k}{100} + \frac{\sqrt{x^2 + d})}{100} - 1} $...
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Can one put absolute values around the first two function terms in the Euler Maclaurin formula for the Riemann zeta function?

I know by the Euler–Maclaurin formula that this is true for the Riemann zeta function: $$\Re(s)>0:\;\;\;\; \zeta (s)=\lim_{k\to \infty } \, \left|\sum _{n=1}^k \frac{1}{n^s}+\frac{1}{(s-1) k^{s-1}}\...
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Asymptotic series for $\ln\binom{x}{k}$ as $k\to\infty$

I was playing around with $\binom{x}{k}$ for fixed $x$ as $k\to\infty$ and realized it could be compared with the Weierstrass factorization of the reciprocal $1/\Gamma(x)$ to get $\binom{x}{k}\sim k^{-...
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Can I use a 'known' Maclaurin series to find the sum of a given series if the series lower bound is not the same?

I'm having a bit of a dilemma right now and I thought I'd ask here. Pretty much, I have a list of known Maclaurin series that I can use on my exam and they greatly help me in solving series and their ...
emcosokic's user avatar
2 votes
1 answer
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Sum of Maclaurin series with centre 0

Sum the power series, $$\sum_{n=0}^\infty\frac{nx^n}{n+9}$$ I found my radius of convergence to be 1 and my interval was $-1<x<1$. For end points I checked and I wasn't too sure if they are ...
Hella Abr's user avatar
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Integral of factional part

Given a function: $$ F(k) = \sum_{k=600}^{999} \left(\left(\pi^2 \right(\frac{k}{1000} + x - 1\left)^2 \right)^{10^{-7}}\right)$$ where $0.1<x<0.35$. The remainder term of the Euler Maclaurin ...
Smith Mayowa's user avatar
4 votes
1 answer
91 views

Why do these two sums define functions that are asymptotically so close in value?

Let $$f_n(\epsilon)=(2n+1)e^{-n(n+1)\epsilon},\qquad Z_\text{odd}(\epsilon)=\sum_{p\ge0}f_{2p+1}(\epsilon),\qquad Z_\text{even}(\epsilon)=\sum_{p\ge0}f_{2p}(\epsilon).$$ (These functions are relevant ...
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2 votes
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440 views

Euler Maclaurin Formula to prove $\zeta(0)=-\frac12$ and $\zeta(-1)=-\frac{1}{12}$

This is a quick question regarding the analytic continuation of the Riemman Zeta function by application of the Euler Maclaurin Formula and the evaluation of $\zeta(0)=-\frac12$ and $\zeta(-1)=-\frac{...
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6 votes
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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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In what case(s) does the Euler-Maclaurin summation method yield the exact evaluation?

So I've been fiddling around with the Euler-Maclaurin summation formula a bit. Here's the formula: $$\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} ...
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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 \\ ...
math student's user avatar
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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(\...
Mathbeginner's user avatar
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1 answer
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How to taylor expand $1/(1- \frac{k}{r})$ where k is some constant

Hi i am doing some integrations and there came across one step that i couldn't understand. The step is as follows: $$\frac{dr}{1 - \displaystyle\frac{k}{r}}=dr\left(1 + \frac{k}{r-k}\right)$$. I don't ...
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How was this simplification done?

\begin{align*} \begin{pmatrix} -2\\ n\end{pmatrix}&=\frac{(-2)(-3)(-4)\dots(-2-n+1)}{n!}\\[5pt] &= (-1)^n\frac{2\times3\times4\times\dots n(n+1)}{n!}\\[5pt] &=(-1)^n(n+1)\\[...
cpt's user avatar
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5 answers
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Taylor series of $e^{-1/x}$

Does the integral converge? $$\int_0^\infty (1-e^{-1/x})\,\Bbb dx$$ Idea: First I tried to expand the $e^{-1/x}$ using the Maclaurin series and evaluated the integral, but it was not a good result and ...
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On determining the constant term in Stirling's approximation

It is well-known that the discrete version of Stirling's formula is as follows $$ \log N!=\left(N+\frac12\right)\log N-N+\log C+\int_N^\infty{P_1(t)\over t}\mathrm dt\tag1 $$ where $P_1(t)=B_1(\{t\})$,...
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Numerical analysis , using Taylor Theorem

On interval [$-\pi ,\pi$ ] , show that $1-\frac{{x}^{2}}{2}\le \cos\left(x\right)$ by using Taylor theorem. I thought that choose ${x}_{0}=0$ and $f(x)=\cos\left(x\right)$. Then, $f\text{'}(x)=-\sin\...
softglance's user avatar
3 votes
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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} + ...
Saïd M's user avatar
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1 answer
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Closed form of infinite sum $\sum_{k=1}^\infty{_{0}F_1}(;2;-kx)$ involving hypergeometric function 0F1

Can we find a closed form for this infinite sum? $$\sum_{k=1}^\infty{_{0}F_1}(;2;-kx)$$ This arises from the problem of finding a closed form for $$\sum_{k=1}^{\infty}\left(-1\right)^{k-1}\frac{\zeta\...
tyobrien's user avatar
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1 answer
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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 ${\displaystyle x}$ ...
HerpDerpington's user avatar
2 votes
1 answer
572 views

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 ...
mhj's user avatar
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3 votes
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How to use Euler-Maclaurin summation to show the following relationship over Gamma function

I encountered this problem that I could not figure out, the first form of the Euler Maclaurin summation is: $$ \sum_{a \leq k \leq b}f(k) = \int_a^b f(x)dx + \frac{f(a)+f(b)}{2} + \sum_{1 \leq i \leq ...
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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 ...
F.sharmin's user avatar
1 vote
1 answer
561 views

The Maclaurin series for $e^{\sin(x)}$ up to the fourth term

I am trying to get the Maclaurin series for $e^{\sin(x)}$ up to the fourth term, which should be: $$e^{\sin(x)}=1+x+\frac{1}{2}x^2-\frac{1}{8}x^4+o(x^4)\tag{1}$$ This is how I go about it: $$e^{\...
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2 answers
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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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0 answers
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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 ...
Hussain-Alqatari's user avatar
2 votes
1 answer
299 views

Infinite sum of cosines as a Gaussian

I have the following function \begin{equation} P(x) = \frac{1}{L} + \frac{2}{L} \sum_{n = 1}^{\infty} \cos\left( \frac{n \pi x}{L} \right) \end{equation} and I try to show that it represents a ...
dancer's user avatar
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2 answers
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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 ...
Agent smith 2.0's user avatar
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1 answer
86 views

evaluation of $\sum_{n=-\infty}^{\infty}e^{-2\pi^{2}z^{2}n^{2}}$ for large z

I want to evaluate the following \begin{equation} \sum_{n=-\infty}^{\infty}e^{-2\pi^{2}z^{2}n^{2}} \end{equation} I know for $z\ll 1$ we can use Euler-Maclaurin formula but in my case z is quite ...
Jason's user avatar
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0 answers
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Determine the specific value of the division of two factorial series

I want to find a specific range of $\alpha$ formula as follows. $$\alpha =\frac{1+\frac{{{x}^{2}}}{3!}+\frac{{{x}^{4}}}{5!}++\cdots+{\frac {x^{2n}}{(2n+1)!}}}{1+\frac{{{x}^{2}}}{2!}+\frac{{{x}^{4}}}{...
Karim's user avatar
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3 votes
1 answer
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Asymptotics (big-O) for power sum

I am trying to prove the following equality, for $\alpha <-1$ and $x \geq 1$ $$\sum_{n \leq x}n^\alpha=\sum_{n=1}^\infty n^\alpha+\mathcal{O}(x^{\alpha+1})$$ and have tried rearranging $$\sum_{n \...
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