Questions regarding the Taylor series expansion of univariate and multivariate functions, including coefficients and bounds on remainders. A special case is also known as the Maclaurin series.

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Difference between the Laurent and Taylor Series.

I have never taken complex analysis, but I am preparing for a GRE for this week end and I am trying to learn a bit about Laurent Series. So far what I get is that the Laurent Series are of form ...
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1answer
395 views

Is the Taylor series comparable to Fourier series and spherical harmonics?

I am currently trying to grasp spherical harmonics and try to digest that we proved that the sine and cosine functions are a basis for the $L^2$ space of the squared-integrable functions. So as far ...
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3answers
342 views

Taylor polynomial of $f(x) = 1/(1+\cos x)$

I'm trying to solve a problem from a previous exam. Unfortunately there is no solution for this problem. So, the problem is: Calculate the Taylor polynommial (degree $4$) in $x_0 = 0$ of the ...
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532 views

Remembering Taylor series

Could anyone suggest a good way of memorizing Taylor series for common functions? I have tried to remember them but never seem to be able to commit them to permanent memory.
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635 views

Prove that the Taylor series converges to $\ln(1+x)$.

Prove the following statement. For $0 \leq x \leq 1$, the Taylor Series, $\displaystyle x - \frac{x^2}{2} + \frac{x^3}{3} - \cdots$ converges to $\ln(1+x)$ Any help will be greatly appreciated! ...
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119 views

Series around $s=1$ for an integral

Consider the function $$F(s)=\int_{1}^{\infty}\frac{\text{Li}(x)}{x^{s+1}}dx$$ where $\text{Li}(x)=\int_2^x \frac{1}{\log t}dt$ is the logarithmic integral. What is the series expansion around $s=1$? ...
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57 views

Analogy to the purpose of Taylor series

I want to know an analogy to the purpose of Taylor series. I did a google search for web and videos : all talks about what Taylor series and examples of it. But no analogies. I am not a math geek and ...
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Other ways to evaluate $\lim_{x \to 0} \frac 1x \left [ \sqrt[3]{\frac{1 - \sqrt{1 - x}}{\sqrt{1 + x} - 1}} - 1\right ]$?

Using the facts that: $$\begin{align} \sqrt{1 + x} &= 1 + x/2 - x^2/8 + \mathcal{o}(x^2)\\ \sqrt{1 - x} &= 1 - x/2 - x^2/8 + \mathcal{o}(x^2)\\ \sqrt[3]{1 + x} &= 1 + x/3 + \mathcal{o}(x) ...
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4answers
51 views

Taylor polynomial of $\frac{1}{2-x}$

Can someone show how to find the Taylor polynomial of $\frac{1}{2-x}$? I tried this: $\frac{1}{2-x}=\frac{1}{1-(x-1)}$ and then use that $ \ T_n(\frac{1}{1-x})=1+x+\dots +x^n.$ But this gives ...
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Show $\forall x>0:\ln(1+x) > x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4}$

I'm reading a proof which aim to show that: $$\forall x>0:\ln(1+x) > x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4}$$ the Taylor expansion of $\ln(1+x)$ is (not by chance): $$x - ...
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85 views

Why does $\arctan(\frac{\tan \theta}{2}) \approx \frac{1}{2 - \theta} - \frac{1}{2 + \theta}$ for small $\theta$?

In answering this question, I needed to show that $\arctan \left( \frac{\tan \theta}{2} \right) \approx \frac{\theta}{2}$ when $\theta$ was small. So, naturally, I computed the first few terms of the ...
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2answers
441 views

How to calculate Taylor expansion of $\cos(\sin x)$

I know that Taylor expansion of $\cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} + O(x^6)$ and that of $\sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - O(x^7)$. But how do I calculte the Taylor Expansion ...
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Counterexample: For real functions existence of all higher order derivatives doesn't imply analycity.

In the lecture we had an example for a function $f: \mathbb R \to \mathbb R$, which is not analytic. We defined, that a function is said to be analytic at some point $x_0$ if a Taylor series expansion ...
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167 views

Taylor series with Fibonacci coefficients

Let $\{a_n\}$ be the Fibonacci numbers given by $a_0=0,a_1=1,a_{n+2}=a_{n+1}+a_n$ for $n\geq 0$. Prove that $f(z)=a_0+a_1z+a_2z^2+\ldots$ is a rational function, and determine which rational ...
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174 views

Taylor series of $\arctan(x+2)$ at $x=\infty$

The simple question is: what is the correct way to calculate the series expansion of $\arctan(x+2)$ at $x=\infty$ without strange (and maybe wrong) tricks? Read further only if you want more details. ...
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106 views

Approximating the logarithm of a Laplace transform

Suppose $X$ is a random variable on $\mathbb R_+$ with finite mean, i.e. $\mathbb E X <+\infty$. Let $F_X(t)$ be its c.d.f. and $\mathcal{L}_X(\cdot)$ its Laplace transform, i.e. ...
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984 views

Showing that the Taylor series of $f(x)=\frac{1}{\sqrt{1+x}}$ converges to $f$ on (I think) the interval (-1,1].

Let $f(x)=\frac{1}{(1+x)^{1/2}}$. Supposed to find the Taylor series $Tf$ of $f$ around $0$, and show that it converges to $f$ on $[-1,1)$, (although I suspect there's a misprint in the book, and that ...
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187 views

Bernoulli and Euler numbers in some known series.

The series for some day to day functions such as $\tan z$ and $\cot z$ involve them. So does the series for $\dfrac{z}{e^z-1}$ and the Euler Maclaurin summation formula. How can it be analitically ...
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877 views

Generalized binomial theorem

Prove that: $$(1+x)^{\alpha}=\sum_{n=0}^{+\infty}{\alpha \choose n} x^n$$ for $x\in[0;1), \alpha \in\mathbb{R}$ based on Taylor's theorem with Lagrange remainder. I don't feel such proofs. ...
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4answers
615 views

Formula for calculating $\sum_{n=0}^{m}nr^n$

I want to know the general formula for $\sum_{n=0}^{m}nr^n$ for some constant r and how it is derived. For example, when r = 2, the formula is given by: $\sum_{n=0}^{m}n2^n = 2(m2^m - 2^m +1)$ ...
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2answers
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Proof of Taylor's series expansion with two terms

I am looking for a simple direct proof of the fact that $$ \frac{\frac{f(x + \Delta x) - f(x)}{\Delta x} -f'(x)}{\Delta x} \stackrel{\Delta x \to 0}{\to} \frac{1}{2}f''(x), $$ or, ...
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765 views

Using Taylor series expansion as a bound

I have a function $f(x)$ that has convergent Taylor series expansion around $x=0$ in the following form: ...
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1k views

Truncation error using Taylor series

How can we use Taylor series to derive the truncation error of the approximation $$f^\prime(x)\approx\frac{f(x+h)-f(x-h)}{2h}$$
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5answers
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Maclaurin polynomial of $\ln(\cos(x))$

I want to write down $\ln(\cos(x))$ Maclaurin polynomial of degree 6. I'm having trouble understanding what I need to do, let alone explain why it's true rigorously. The known expansions of ...
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76 views

If f ' = 0, then f is constant?

I'm a little confused. After finishing the online multi-variable calculus course from the MIT OCW offerings (I wanted to brush up on the subject more concretely, after my Analysis II course), I ...
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Showing that the integral remainder of the Taylor expansion of $f(x)=-\log(1-x)$ goes to $0$

Let $|x|<1$. Define $R_n(x):=\int_{0}^{x}\frac{(x-t)^{n-1}}{(1-t)^n}dt$. How do we prove that $\lim_{n\to \infty}R_n(x)=0$? This is actually the integral remainder of the Taylor expansion of the ...
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Remainder term in Taylor's theorem

I'm trying to understand the remainder in Taylor's theorem from this source: https://proofwiki.org/wiki/Taylor%27s_Theorem/One_Variable/Integral_Version I don't understand the very last parts of ...
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66 views

Intuition behind Taylor/Maclaurin Series

** This is a different question than Intuition explanation of taylor expansion? ** I understand some of the intuition behind a Taylor/Maclaurin expansion. More specifically, I understand that adding ...
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Integration by expansion

Consider the integral \begin{equation} I(x)= \frac{1}{\pi} \int^{\pi}_{0} \sin(x\sin t) \,dt \end{equation} show that \begin{equation} I(x)= \frac{2x}{\pi} +O(x^{3}) \end{equation} as ...
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161 views

Why don't taylor series represent the entire function?

Say, I have a continuos function that is infinitely differentiate on the interval $I$. It can then be written as a taylor series. However, taylor series aren't always completely equal to the function ...
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How to show that $e^{x+y} = e^x e^y$ by series expansion [duplicate]

I know that $e^xe^y=e^{x+y}$ but I want to show it by expanding the exponentials in MacLaurin Series. $$ \left(\sum_{n=0}^{\infty} \frac{x^n}{n!}\right) \left(\sum_{m=0}^{\infty} ...
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758 views

Taylor series convergence for $e^{-1/x^2}$

Consider the Taylor series for $e^{-1/x^2}$ around $0$: $$e^{-1/x^2}=1-\dfrac{1}{x^2}+\dfrac{1}{2!x^4}-\dfrac{1}{3!x^6}+\ldots$$ For which $x$ does the series on the right converge to $e^{-1/x^2}$?
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Find the Maclaurin series for $\cos(2x)$ using the series for $\sin(2x) $.

I know that $$\sin(2x)= 2x - \frac{8x^3}{3!} + \frac{32x^5}{5!} - \frac{128x^7}{7!} + \cdots $$ $$\sin(2x)= \sum_{n=0}^\infty (-1)^n {2^{2n+1}x^{2n+1} \over (2n+1)!}$$ But I don't see how I can use ...
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448 views

Sine taylor series

I'm pretty convinced that the Taylor Series (or better: Maclaurin Series): $$\sin{x} = x -\frac{x^3}{3!} + \frac{x^5}{5!} - \cdots$$ Is exactly equal the sine function at $x=0$ I'm also pretty sure ...
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302 views

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

I'm currently stuck on the question regarding the Maclaurin series for $e^x +2e^{-x}$ I've found that the power series representation for it is $$\sum_{n=0}^\infty \dfrac{x^n + 2(-x)^n}{n!}$$ ...
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193 views

How does a taylor series of a binomial function equals a trigonometric function? [closed]

Any proof or derivation for the sinx and cosx function would be help. Image taken from http://en.wikipedia.org/wiki/Taylor_series
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739 views

Infinite (Taylor) Series

What is the general term for a Taylor series of $\sin(x)$ centered at $\pi/4$? It should be $(-1)^{[??]} \times \sqrt{2}/2 \times \frac{(x-\pi/4)^n}{n!}$ What power is $(-1)$ supposed to be raised ...
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780 views

Find the taylor series for $\cos x$ and indicate why it converges to $\cos x$ for all $x\in \Bbb R$.

Find the taylor series for $\cos x$ and indicate why it converges to $\cos x$ for all $x\in \Bbb R$. I've posted my own proof, I hope it is correct :)
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What exactly are the “higher order terms” (H.O.T.) in Taylor series expansion in multivariable case?

I know the Taylor series expansion in single variable case: $$ f(x) = f(x_0) + f'(x_0)(x - x_0) + \frac{1}{2}f''(x_0)(x-x_0)^2 + \frac{1}{3!}f^{(3)}(x_0)(x-x_0)^3 + \frac{1}{4!}f^{(4)}(x_0)(x-x_0)^4 ...
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Exercise about MacLaurin's polynomial and small-o

In class the professor wrote the following limit: $\lim_{x\to 0} \frac{\sinh^2 (x) -x^2}{x^4}$ So he "expanded" (sorry for my English) the MacLaurin's formula for $\sinh x$ up to the 3rd power, and ...
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First four Taylor Series Expansions

I'm supposed to write the first four taylor series expansions of $f(x=0)$ using: one term, two terms, three terms, four terms This is the function: $$f(x) = x^3 - 2x^2 + 2x - 3$$ Should I be using ...
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1answer
118 views

Taylor polynomial for $f(x)=x^{1/7}$ about $a=1$

Here is the problem: (a) Determine the Taylor polynomial $T_2(x)$ of degree $2$ for the function $f(x)=x^{1/7}$ centered at $a=1$. (b) Suppose we were to use the approximation $f(x) \approx ...
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318 views

Laurent series for an infinite sum of functions

I have an infinite sum of analytic functions that is guaranteed to converge for every $x$, except for $x=0$: \begin{equation} g(x) = \sum_{n=1}^\infty f_n (x) \end{equation} I want to expand the ...
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Taylor's Theorem For Error Approximation

I'm trying to evaluate a function $f(t)$ with a given $t$ value to within 10$^{-5}$. So, if I use Taylor's Theorem : $f(a)+f′(a)1!(x−a)+f′′(a)2!(x−a)2+⋯+f(n)(a)n!(x−a)n.$ Would my $t$ value = $a, ...
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Finding the $n$th Taylor coefficient of $g(z)=\frac{z}{(z-b)^2}$ centered at $a$ (where $a=2-\sqrt{3}$ and $b=2+\sqrt{3}$?

I've introduced $a$ and $b$ in order to simplify the notation : $a=2-\sqrt{3}$ and $b=2+\sqrt{3}$. I'm trying to compute the Taylor Series for $g(z)=\frac{z}{(z-b)^2}$ centered at $a$. I denote the ...
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2answers
38 views

Find series expansion of 1/cosx

Find the series expansion of 1/cosx from basic series expansions. I tried to find 1/cosx from the expansion of cosx but was unsure how to continue. When I found 1/cosx from the basic formula for ...
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1answer
37 views

Radius of convergence of $(1+x)^p$

Problem: Show that $(1+x)^p$ converges everywhere for $p \in \mathbb{N}$, and for $|x| < 1$ otherwise. My work: I think that if $p \in \mathbb{N}$ then the Taylor series will just be a polynomial ...
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2answers
94 views

How to find a closed form for the derivatives of $F(x)=\frac1x\int_0^x\frac{1-\cos t}{t^2}\,dt,$ $F(0)=\frac12$?

I have been given the function $$F(x)=\frac1x\int_0^x\frac{1-\cos t}{t^2}\,{\rm d}t$$ for $x\ne 0,$ $F(0)=\frac12,$ and charged with finding a Taylor polynomial for $F(x)$ differing from $F$ by no ...
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151 views

On the hessian matrix and relative minima

I'm asked to prove the following statement: Let $\mathbb{A} \subseteq \mathbb{R}^n$ an open subset and $f: \mathbb{A} \to \mathbb{R} / f \in \mathbb{C}^3 \wedge (P \in \mathbb{A} : \nabla f(P)=0)$. ...
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512 views

Express $\sin nx$ and $\cos nx$ in terms of $\sin x$ and $\cos x$ respectively

What are the expansions of $\sin nx $ and $\cos nx$ in terms of $\sin x$ and $\cos x$ respectively? (here $n \in \mathbb N$). Maybe this is solved problem or there is new technique to ...