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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16 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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39 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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26 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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30 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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39 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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Questions About Complex Rotation

This animation shows that the infinite sum of vector tips representing the Maclaurin Series of $e^{i\theta}$ lies on the unit circle in the complex plane for any value of $\theta$. The first vector ...
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1answer
39 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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20 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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67 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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Trapezoidal quadrature convergence for Holder continuous functions

Suppose $f \in C^{k, \alpha}(a,b)$, with $f^{(j)}(a) = f^{(j)}(b)$ for $j = 0,1,\ldots, k$. We have at least that the error of the trapezoidal rule is $$ |I[f] - T_{h}[f]| \le h^{k}\int_{a}^{b} \tilde{...
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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}}}{...
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114 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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49 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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33 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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Maclaurin series, the definition [closed]

Hey I will be glad if someone can explain to me the reasoning behind the domains of Maclaurin series. For example: In the "text book" it says: $$ {1\over 1-Z} = 1 + Z +Z^2 + Z^3+\dots + Z^n + \dots = \...
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37 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
45 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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90 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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Are these steps correct to find Taylor series?

Ex.: Find the Taylor series (3rd grade) $f(x)=sinx / (2+3x)$ @ (0,0) 1) I will find $f'(x), f''(x) and f'''(x)$ 2) Then $f(0), f'(0), f''(0) and f'''(0)$ 3) Use Maclaurin series is that ok ? ...
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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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225 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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244 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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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
43 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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101 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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93 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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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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62 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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53 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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60 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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60 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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98 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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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}{...
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102 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_{...
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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 ? $$\int_{0}^{1}\bigg\{\frac{1}{x^s}\bigg\}dx\,$$
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160 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\...
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166 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 ...
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190 views

Numerical integration of sharply peaking function: $\int_0^{1000} \exp(1000\sin(x)/x)\,{\rm d}x $

$$ \int_0^{1000} \exp(1000\sin(x)/x)\,{\rm d}x$$ Solve this integral of a sharply peaked function without a calculator. I was told to not do an expansion of the sharp function, but of its gently ...
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1answer
168 views

Bounds for Periodic Bernoulli Polynomials via Fourier Series

I am looking at the following result here on page 121. The Fourier series of the $p$'th Periodic Bernoulli polynomial $\mathcal{P}_p(x) := \mathcal{B}_p(\{x\})$ ($p$'th Bernoulli polynomial evaluated ...
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1answer
525 views

the remainder of Maclaurin Series

I don't know how to prove/disprove this: $f(x)$ is a differentiable function from any degree in R. let $R_(n)$ be the remainder of Maclaurin Series of the function f(x). I need to prove or disprove:...
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1answer
35 views

Maclaurin polynomial error bounds [closed]

I need some help with my Calculus II Maclaurin polynomial error bounds. $Mn(x)$ is the $n^{th}$ Maclaurin polynomial for $f(x) = e^x$. I need to use the error bound formula to determine a value of $...
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1answer
181 views

How to integrate the “fractional part”/sawtooth function?

Let us define $\{t\} = t - \lfloor t \rfloor$ this is also sometimes referred to as frac$(t)$. With this in mind how would I calculate $$I := \int_a^b f(\{t\}) dt$$ for some function $f$? I ask ...
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6th Degree Polynomial and Chebyshev minmax

I have to find the 6th degree polynomial for the function $f(x)=xe^x$. After which the use of the Chebyshev min-max approach I have to use the list grade polynomial approach with respect to the fault ...
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86 views

How to find value of arctan using Maclaurin

I need to find $\arctan(3)$ and before this, I'm asked to find the Maclaurin series for $\arctan(x)$, so I know they must be related. But $\arctan(x)$ is defined with $x \in [-1,1]$ so what can I do?
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23 views

Stuck evaluating Taylor series

We are told to evaluate the Taylor series of $f(z) = \dfrac{z+i}{z-3}$ around $z = 4i$ and that the final answer must be in summation notation My work: (I have skipped steps for simplicity) $f(z) = \...
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2answers
58 views

Find The Maclaurin series of $5x^2e^{-8x}$

I'm having trouble finding the Maclaurin series of $5x^2e^{-8x}$ and determining the coefficients. the initial $e^{-8x}$ is simple enough but after that I get confused, our teacher rushed through ...
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1answer
4k views

Finding the MacLaurin polynomial of degree $7$? [closed]

The question reads: let $F(x)=\int_0^x \sin(6t^2) \, dt$ 1) Find the MacLaurin polynomial of degree $7$ for $F(x)$. 2) Use this polynomial to estimate the value of $\int_0^{0.75} \sin(6x^2)\,dx$. ...
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43 views

How to find the maclaurin expansion for an implicit trig function? $y=\sin(x+y)$

I'm currently creating a function for my final maths assignment, which I'm seeking to find a maclaurin expansion for. The specific function is: $$ f(x,y)=\sin((2π/1.17)x+(π/2-1.047)y) $$ I know that ...
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2answers
301 views

Finding the limit of $\left(\sum\limits_{k=2}^n\frac1{k\log k}\right)-\left(\log \log n\right)$ [duplicate]

There is this well known limit: $$\lim_n \sum_{k=1}^n \frac 1k -\log n=\gamma$$ Where $\log$ is the natural logarithm and $\gamma$ is Euler constant. I was wondering if my similar situation yelds ...
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848 views

Difference b/w Taylor and Maclaurin Series [duplicate]

How is Maclaurin Series different from Taylor Series? With a little bit of surfing, I figured out that Maclaurin series is an approximation about the point $0$. Does that mean that Maclaurin series ...