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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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Integrating the remainder of EMSF for $\sum_{x=1}^n\ln x$

The remainder term is determined by the following formula $$R_{m,n} = \frac{(-1)^{(p+1)}}{p!}\int_m^nf^{(p)}(x)P_{(p)}(x)dx$$ Let $m = 1, p = 4, f(x) = \ln x, f^{4}(x) = \frac{-6}{x^4}$ the formula ...
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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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Convergence in Euler-Maclaurin

In the Euler-Maclaurin formula, we get an asymptotic approximation with the error term arbitrary small. However, I am confused about the convergence. The problem is that the constants in the terms ...
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0answers
49 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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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 $f$ ) ...
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2answers
37 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
54 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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42 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
82 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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48 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}{...
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2answers
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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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2answers
75 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 ? $$\int_{0}^{1}\bigg\{\frac{1}{x^s}\bigg\}dx\,$$
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1answer
107 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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31 views

On the Puiseux series of divergent zeta function for $0 < \Re(s)< 1$

Let $s$ be a complex number such that 0 < $\Re(s) < 1$ and $\zeta(s)$ be the analytic continuation of zeta function on the strip 0 < $\Re(s) < 1$. Then by applying the Euler-Maclaurin ...
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1answer
132 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
107 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
96 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
28 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
130 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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59 views

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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1answer
79 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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1answer
22 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
46 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
2k 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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36 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
181 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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2answers
408 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 ...
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2answers
84 views

Maclaurin series for $x e^{x^2-1}$

i have done everything to that step$$xe^{x^2-1}=f(x)=\frac 1 e \sum_{n=0}^\infty \frac{x^{2n+1}}{n!},$$ but I need $f^{(101)}(0)$ so I know that I need to find $2n+1=101$ which is $n=50$. But I don't ...
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2answers
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Proof that $e^x = 1 + x +\frac{x^2}{2!}+\frac{x^3}{3!}+\ldots$

I'm very new to Taylor and/or Maclaurin Series, and the main thing that I understand is actually not rigorous at all, and is that: Taylor series involves expressing a function as the sum of an ...
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5answers
122 views

Maclurin series for $e^{\sqrt{x^2+1}}$

i am giving $e^{\sqrt{x^2+1}}$ and asked to find the Maclaurin series for this term. Here is my solution: let $u=\sqrt{x^2+1}$, and given that we know that Maclaurin series for $e^x= 1+x+\frac{x^2}{...
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0answers
241 views

Use Maclaurin series of a function to determine another

The prompt is to determine Maclaurin series of the function $f(x) = -x^5ln(1-2x^3)$ using the Maclaurin series for $g(x) = ln(x+1)$ also to find for what value of x the series f(x) converges, and ...
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3answers
116 views

Finding Maclaurin series of a function

The prompt is to find the Maclaurin series of the function $$f(x) = xe^{x^2-1}$$ and evaluate the 100th derivative. At what value of x does this series converge? We know that $$e^x = \sum \frac{x^n}{...
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163 views

Chain rule: derivative vs integral

What is the difference between the chain rule in a derivative versus the application of the chain rule in an integral? For example, in the case of derivative of $\ln(1+4x)$? I've done so many ...
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3answers
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How can $i^i = e^{-\pi/2}$ !!

I was asked a homework question: find $i^i$. The solution provided was as follows: Let $A = i^i$. $\log A = i \log i$. Now, $\log i = \log e^{i\pi/2} = \frac{i\pi}{2}$. So, $\log A = -\frac{\pi}{2}...
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2answers
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Finding antiderivative by Maclaurin series

I want to find $\int p(x) dx$ where: $$p(x)= \frac{e^{\frac{-x^2}{2}}}{\sqrt{2\pi}}.$$ This function can not be manually computed so I am using Maclaurin series to find the antiderivative. I believe ...
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1answer
34 views

Find the value of the sum with accuracy up to $o(1)$.

Consider the sum $S_n = 1 + \frac13 + \frac15 + ... + \frac{1}{2n-1}$. How do we find its sum with accuracy up to $o(1)$?
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1answer
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Maclaurin representation of a function $f(x) = sin(\pi x)$

$$ f(x) = sin(\pi x) $$ Find the correspondent Maclaurin series for the function $f(x)$. Here's my approach: Since Maclaurin series is defined as: $$ \sum_{n=0}^{\infty}\frac{f^{(n)}(0)}{n!}x^n $$ ...
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4answers
443 views

Maclaurin series of $f(g(x))$

I was doing some exercise about Maclaurin expansion when I notice something, I used to remember the series formula of some common functions with $x$ as argument, but when I had to calculate the ...
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0answers
57 views

Euler-Maclaurin summation formula for piecewise continuous functions

The Euler-Maclaurin formula is essentially like this: Suppose that $f$ is continuously differentiable in an interval $[a,b]$ with $a < b$ integers. Then $$ \sum_{a < n \leq b} f(n) = \int_{a}^b ...
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2answers
164 views

Prove an equation holds in series: $\sum\limits_{x=0}^{\infty} xe^{-\lambda}\frac{\lambda^x}{x!}=\lambda$

Let $\lambda\in \mathbb{R}$ be a constant. Prove $\sum\limits_{x=0}^{\infty} xe^{-\lambda}\frac{\lambda^x}{x!}=\lambda$. I have already proved $\sum\limits_{x=0}^{\infty}e^{-\lambda}\frac{\lambda^x}{...
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1answer
80 views

How fast does $\lim_{x\to 1^-} \sum_{k=0}^\infty \left( x^{k^2}-x^{(k+\alpha)^2}\right)$ go to $\alpha$?

In this question: $\lim_{x\to 1^-} \sum_{k=0}^\infty \left( x^{k^2}-x^{(k+\alpha)^2}\right)$ it is established that $$\lim_{x\to 1^-} \sum_{k=0}^\infty \left( x^{k^2}-x^{(k+\alpha)^2}\right) = \...
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1answer
48 views

Develop in series Mac-Laurin the function $f(x)={\frac{x^2}{1-x}}$

I just want to know if what I did was good because I did not fully understood this theorem. I have this function: $$f(x)={\frac{x^2}{1-x}}$$ And I have to develop her in series Mac Laurin. Firstly I'...
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2answers
77 views

Summing a series of integrals

I asked this question on Mathoverflow, but it was off-topic there (though it is related to my research...) and I was told to ask it here. I have a series of integrals I would like to sum, but I don't ...
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0answers
889 views

Maclaurin polynomial of order 3? Order vs. Degree

I am doing some homework and came across a problem that asks: Find the Maclaurin polynomial of order 3 for f(x) = e^(-4x) When did some searching online, all searches came up as "...maclaurin ...
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1answer
80 views

developing to maclaurin Series $f(x)=\frac{2x+3}{x^2 -4x+5}$ on $x=2$

$$f(x)=\frac{2x+3}{x^2 -4x+5}$$ on $x=2$. My solution: $t=x-2 $ => $x=t+2$ , we get: $f(t)=\frac{2t+7}{t^2+1}$ on $t=0$. then: $(2t+7)\sum_{n=0}^{\infty } {(-t^2)^n} = (2t+7)\sum_{n=0}^{\infty }{(-...
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1answer
248 views

Stirling's approximation from Euler-Maclaurin formula

I try to derive Stirling's approximation from Euler-Maclaurin formula with form: $$\sum_{x=m}^nf(x)=\int_m^n{f(x)dx}+\frac{f(n)+f(m)}{2}+\sum_{k=2}^p{\frac{(-1)^kB_k}{k!}[f^{(k-1)}(n)-f^{(k-1)}(m)]+\...
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2answers
127 views

Reindexing Exponential Generating Function

I have an exponential generating function, and I need to double check what the teacher said, because I'm having trouble coming to the same result. Also, I need to verify what I am coming up with, and ...
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0answers
63 views

Use the Euler–Maclaurin formula to find $P_{k+1}$ where $P_{k+1}(n)=\sum_{j=1}^n j^k$ for $2\le k \le 10$

I am not familiar with the Euler–Maclaurin formula. Any idea how to Use the Euler–Maclaurin formula to find $P_{k+1}$ where $P_{k+1}(n)=\sum_{j=1}^n j^k$ for $2\le k \le 10$? Thanks!
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1answer
93 views

How to find the sum: $ \sum_{i = 0}^n i^{1/5} $

Given the sum: $$ \sum_{i = 0}^n i^{1/5} $$ How to find $A$ in: $$ \sum_{i = 0}^n i^{1/5} = A + O(\frac1{n^6}) $$ I tried to use Euler–Maclaurin formula and obtained numbers that confused me?
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1answer
46 views

Is there a good way to compute this integral?

Sometimes questions like: "How many digits does 2015! have?" become quite trendy. The most reasonable approach would probably use Stirling's formula. However doing this in the same way over and over ...