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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31 views

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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1answer
30 views

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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1answer
33 views

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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13 views

$n$-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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2answers
48 views

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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0answers
22 views

Midpoint rule applied to $x^\alpha f(x)$

I want to approximate $$ \int_0^1 x^\alpha f(x)dx $$ by means of the mid-point rule, where $\alpha > -1$ and $f(x)$ is a $\mathcal C^\infty$ function on $[0, 1]$, i.e., if $h = 1 / N$, $x_{n+1/2} \...
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1answer
30 views

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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42 views

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)\\[...
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4answers
104 views

Taylor series of $e^{-{1}/{x}}$

Does the integral converge? \begin{align*} \int_0^\infty (1-e^{-{1}/{x}})dx \end{align*} Idea: First I tried to expand the $e^{-{1}/{x}}$ using the Maclaurin series and evaluated the integral but it ...
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1answer
45 views

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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1answer
42 views

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\...
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97 views

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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1answer
54 views

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\...
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1answer
25 views

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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50 views

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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1answer
93 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 ...
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1answer
219 views

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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22 views

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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1answer
129 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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2answers
33 views

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.stack.imgur.com/bUhxk.jpg
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35 views

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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1answer
44 views

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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1answer
71 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 ...
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2answers
22 views

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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1answer
71 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 ...
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27 views

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}}}{...
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1answer
143 views

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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1answer
798 views

Deriving the Maclaurin Series for sin(x)/x

When finding the Maclaurin series representation for sin(x)/x, I decided to multiply the Maclaurin series for each individual function first. The Maclaurin series for sin(x) is: $\sum_{n=0 }^{\infty}\...
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1answer
45 views

Verifying Big O of remainder term when deriving Stirling's approximation formula

I am trying to verify the following statement mentioned in an wikipedia article (https://en.wikipedia.org/wiki/Stirling%27s_approximation#Derivation), the statement is used to derive the stirling's ...
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0answers
47 views

A way to approximate or reexpress the Bernoulli number containing sum in the Euler-Maclaurin approximation of the Riemann-zeta function?

Please examine the first (and second) equations here. I would like to find a way to express the sum containing even Bernoulli numbers as a closed form function (evaluate it) or approximation of it ...
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1answer
54 views

Do Taylor series model their functions to 100% accuracy?

Do Taylor series model their functions to 100% accuracy? For example, is $sin(x)$ equal to $\lim_{n\rightarrow\infty}T_n(x)$ for all x?
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1answer
684 views

Absolute value of a Maclaurin series

I am trying to find out the absolute value of a Maclaurin power series of the below type: $f(x) = a_0 + a_1 x+a_2 x^2+ \dots + a_n x^n$, where $x$ is a complex number. I am interested to know the ...
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2answers
41 views

10th derivative by MacLaurin

I'm sorry for my language. English is not my first language. I'm trying to find $$f^{(10)}(0)\;\;\text{when}\;\;f(x)=\frac{1}{2+x}$$ by using MacLaurin. The answer is: $$f^{(10)}(0)=\frac{10!}{2^{11}}...
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2answers
407 views

Derive taylor series of $e^{\sin(x)}$ in two different ways

I need to find the Taylor series of $e^{\sin(x)}$ up to $x^4$ in two different ways. First I derived it by calculating the derivatives of the function, and I found the answer $P_4(x) = 1+x+ \frac{x^2}{...
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2answers
261 views

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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3answers
87 views

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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2answers
88 views

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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0answers
139 views

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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0answers
124 views

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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0answers
52 views

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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0answers
88 views

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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2answers
101 views

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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1answer
66 views

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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2answers
70 views

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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1answer
113 views

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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0answers
58 views

Convergence of sum to integral

I would like to estimate the absolute value of the following difference $$ \Delta(L) = \sum_{\alpha=-L+1}^L \frac{1}{1+2 L} e^{i t \sec^2\left(\pi\frac{\alpha - 1/2}{2 L+1}\right)} - \int_{-\frac{1}{...
4
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2answers
107 views

Taylor series of functions

Consider the Taylor series of the function $$\frac{2e^x}{e^{2x}+1} = \sum_{n=0}^{\infty} \frac{E_n}{n!} x^n$$ Prove that $E_0 = 1, E_{2n-1} = 0$ and, for $n \ge 1$, $$E_{2n} = - \sum_{l=0}^{n-1} C_{...
2
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2answers
119 views

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 ? $$\int_{0}^{1}\bigg\{\frac{1}{x^s}\bigg\}dx\,$$
2
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1answer
204 views

Are the partial sums for $\sum_{n=1}^{\infty}\sin(n^a)$ bounded for $a\geq1$ and unbounded for $0<a<1$?

I know that the partial sums of $$\sum_{n=1}^{\infty}\sin(n)$$ are bounded between $\frac{\cos\left(\frac{1}{2}\right)-1}{2\sin\left(\frac{1}{2} \right)}$ and $\frac{1+\cos\left(\frac{1}{2} \right)}{2\...
5
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1answer
271 views

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 ...