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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Proof that taylor series converges to function using taylors inequality

I would like to proof that the function $f(x)=\frac{1}{\sqrt{1-x}}$ converges to its Maclaurin series $$Tf(x;0)=1+\sum_{n=1}^\infty{\frac{(2n)!}{(2^n n!)^2}}x^n$$ for $0<x<1$ by using taylors ...
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Find the quadratic Taylor polynomial of erf about a=0?

The answers say that $P_3(x) = 1 + \frac{1}{2} x^2 $ I understand that this is the sum of the first four terms, however I don't know how they calculated this. I know that the formula for the ...
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1answer
26 views

Taylor polynomials: remainder formula for expansion around $\infty$.

By definition of Taylor polynomials, we have $$f(x)=f(x_0)+f'(x_0)(x-x_0)+...+\frac{f^{(n)}(x_0)}{n!}(x-x_0)^n+R_n(x,x_0)$$ where $R_n$ is the $n-$th remainder . Let $U(x_0)$ the neighborhood of ...
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1answer
45 views

Taylor expansion of fraction

I am trying to Taylor expand the function $f(x) = \frac{x}{x+3}\frac{1}{x-2}$ aound the point $x_0 = 2$. Clearly, the last factor explodes around this point, so I will try and expand that term. ...
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Proving the inequality $2^{-1+\frac{1}{n}}\left(x+1\right)\leq\left(x^{n}+1\right)^{\frac{1}{n}}$ for $n\in\mathbb{N} $ and $x>0$

Prove that for all $n\in\mathbb{N}$ and $x>0$, $$2^{-1+\frac{1}{n}}\left(x+1\right)\leq\left(x^{n}+1\right)^{\frac{1}{n}}$$ The last class was about Taylor polynomial of functions, so I ...
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42 views

Taylor series of $e^{i \sin z}$

How can I find the Taylor series of at $z=0$ (where $z$ complex ) of: $e^{i \sin z}$? What I wrote is: $$e^{i \sin z}= \sum_0^\infty \frac{(i \sin z)^n}{n!}$$ Is that right? And how it can be more ...
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52 views

Taylor series and Maclaurin series problems

I'm currently working on these two problems, and I'm getting really confused with them. Can someone walk me through them? Find the Maclaurin Series for $f(x)=\cos\left(\sqrt x\right)$ and use ...
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linear term of $\sqrt{ 1 - A \exp^{2iwt} - B exp^{2iwt}}$ does it make nonsense to analysis of dynamics?

I have the following problem: I am building a Lagrange Euler equation near the position where some vector component equals to 1, while others no. I do the following substitutions: $$\vec x = ...
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24 views

First order approximation?

Why is always the first order approximation which is used. I don't see the benefit of an inaccurate guess. Computational wize i see the benefit, as it doesn't take that many differentiation to do, but ...
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Why doesn't the Taylor series always converge?

The Taylor expansion itself can be derived from mean value theorems which themselves are valid over the entire domain of the function. Then why doesn't the Taylor series converge over the entire ...
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1answer
29 views

It is possible to demonstrate the taylor's formula (Peano) with the taylor's formula (Lagrange)?

I wonder if it is possible demonstrate the taylor's formula in the peano form of the reminder with the taylor's formula in the lagrange formula of the reminder. In symbols: ...
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9 views

Bounding the error term in Taylor's theorem for multivariate function

I have a multivariate function f. I am expanding it according to the Taylor's expansion. I am evaluating it at two points x and y near a fixed point q. I need to evaluate the value of the difference ...
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69 views

Approximate $\ln2$ using the Maclaurin series expansion of $\ln\cos x$

Until now I have that $\ln\cos x\approx-\dfrac{x^2}{2}-\dfrac{x^4}{12}$. Since $\cos^{-1}2$ does not exist I do not know what value of $x$ to take. I suppose I need to play around with the original ...
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1answer
43 views

Are there any entire functions that have a finite number of non-zero terms in their Taylor expansions?

Besides polynomials, of course. I am pretty sure the answer will be "no", but can we actually prove this? Thanks,
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51 views

How do I derive the Maclaurin series for $\tanh(x)$?

I've thought of doing it by writing $\tanh(x)$ as $(1-e^{-2x})/(1+e^{-2x})$ and then using the Maclaurin series for $e^{x}$ or just as $\sinh(x)/\cosh(x)$ and using the Maclaurin series for $\sinh(x)$ ...
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49 views

Help with a Taylor expansion

I want to find the Taylor polynomial $T_{n,f,0} (x)$ of the function: $f(x)=\displaystyle\int_{0}^{x} e^{-t^2}dt$ It cannot be done using the classic calculations, but I may use the theorem of ...
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19 views

On a Lagrange remainder application.

Suppose I have $f(x)$ where $f$ is three times continuously differentiable. Then from the Lagrange form of the remainder $ R_k(x) = \frac{f^{(k+1)}(c)}{(k+1)!} (x-x_0)^{k+1} $ I have $$f(x) - ...
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2answers
34 views

Maclaurin Series of $\ln(2-e^{-x}) $

I tried to solve this by using the series for $e^{-x}$ and $\ln(1+u)$ $$e^{-x}=1-x+\frac{x^2}{2}-\frac{x^3}{6}+...\\ ...
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2answers
44 views

Quick Method to Calculate the Maclaurin Series of $\frac{1}{\sqrt{\cos{x}}} $

I am supposed to calculate the maclaurin series for $\frac{1}{\sqrt{\cos{x}}} $ but I can't seem to figure out an efficient way to go about doing this.
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The power series $\sum_{n\geq 1} \frac{x^{n}}{n(2n-1)}$ with $2$nd Taylor polynomial and Taylor series. [Solved]

I have been a fool not noticing it earlier. Instead of deleting this thread I have chosen to put the short solutions of this problem. This thread is closed. Consider the series $$\sum_{n\geq 1} ...
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Maclaurin Series of $\frac{2x}{e^{2x}-1}$ [duplicate]

Calculate the Maclaurin series of $$\frac{2x}{e^{2x}-1} $$ I've tried to calculate it but the series $\frac{1}{e^{2x}-1}$ divides by 0 when x is equal to 0
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Which is greater ? Sum of odd power terms or even power terms in the exponential Taylor series?

I came across this question, in a book. Define $f(x) = \sum_{n=0}^{\infty} \frac{x^{2n+1}}{{(2n+1)}!} $ and $ g(x) = \sum_{n=0}^{\infty} \frac{x^{2n}}{{(2n)}!} $, where x is a real number. Then, ...
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1answer
63 views

Composition of Pseudodifferential Operators - Remainder term of Asymptotic Expansion

first off this is my first time posting here so I am only learning how to format questions. Please bear with me. I am trying to prove the asymptotic expansion for the symbol of the composition of ...
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Proof of a step of a lemma on the asymptotics of maximum likelihood where a Taylor expansion is used. (crosspost from crossvalidated).

I have asked this question on crossvalidated here and I am still unsure on the answer. I attempt a cross-post (most of the times this proves very useful). I copy the question below: I am trying to ...
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1answer
59 views

Taylor Expansion of tensor moved by a flow.

I am reading Peter Petersen's notes on manifold theory and he introduces Lie Derivatives in the following way. "Let $X$ be a vector field and $F^t$ the corresponding locally defined flow on a smooth ...
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2answers
66 views

Taylor series of $[\log(1-z)]^2 $

I'm having some trouble proving that the Taylor series about the origin of the function $[Log(1-z)]^2$ to be $$\sum_{n=1}^\infty \frac{2H_n}{n+1}z^{n+1}$$ where $$H_n = \sum_{j=1}^n \frac{1}{j}$$ So ...
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1answer
13 views

Taylor expansion at two different points.

My question is stated in the image. I want to prove that every coefficient is same for all terms. But it is not easy. Can you give me some direct proof?
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46 views

Find $\lim\limits_{x \to 0} \frac{(1+3x)^{1/3}-\sin(x)-1}{1-\cos(x)}$

I would like to find using Taylor series : $$\lim\limits_{x \to 0} \frac{(1+3x)^{1/3}-\sin(x)-1}{1-\cos(x)}$$ So I compute the taylor series of the terms at the order $1$ : $(1+3x)^{1/3}=1+x+o(x)$ ...
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28 views

Taylor expansion need help understanding.

I am at the moment reading a paper (SURF) and trying to understand what is happening here and how the things works as it does.... a non maximum supression is performed on the scale space ...
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3answers
84 views

The maclaurin series $ f(x) =\frac {x^3} {2+ x^2}$

I know we have exams today and I am doing practise since our lecture; said we need to review our Maclaurin series and I found this question and I wanted to know how one would approach it. Find the ...
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Taylor series of a twice-differentiable scalar function

I've come across this passage somewhere on wikipedia: If $f(t,x)$ is a twice-differentiable scalar function, its expansion in a Taylor series is $$df = \dfrac{\partial f}{\partial t}dt + ...
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The Taylor series of the squared logarithm [duplicate]

Prove that the Taylor series about the origin of the function $[\log(1-z)]^2$ is given by $$\sum_{n=1}^{\infty} \frac{2H_{n}}{n+1} z^{n+1}$$ where $$H_{n} = \sum_{j=1}^{n}\frac{1}{j}$$ is ...
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38 views

Analytic continuation of ln(z) counterclockwise about the unit circle,

We write ln(z) as ln(1+z-1) = ln(1+(z-1)) to utilize the familiar expansion that is: (z-1) - (z-1)^2 / 2 + ... which converges for |z-1| < 1, i.e., we get convergence of ln(z) in an open Taylor ...
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1answer
48 views

What is the Taylor series of $e^x$ centred at $3$?

$$ \sum_{k=0}^n \frac{e^3}{n!}(x-3)^n $$ This is my answer - is it correct?
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1answer
19 views

Taylor series at $a = 0$

If we are given a power series with $a = 0$ and it converges to $f$ in some interval around $a$, then the power series is the taylor series to $f$. But what is the taylor polynomial of $f$ to some ...
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1answer
33 views

Approximation formula for third derivative, is my approach right?

Derive by using Taylor approximation up to 4th degree (in $h$) of $f$ in $x_0 \pm h$, $x_0\pm 2h$ at $x_0$, an formula for approximation of $f'''(x_0)$ with an error term of order $h^2$. Could ...
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1answer
38 views

Complex number, series representation

Show that for any finite value of $z$ $$e^z=e+e\sum_{n=1}^\infty \frac{(z-1)^n}{n!}$$ For $z=1$ $$f(z)=f(z_0)+\sum f^{(n)}(z_0)\frac{(z-z_0)^n}{n!}$$ equality is checked, but I do not know how to ...
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0answers
11 views

Taylor series for a multivariable function

We know the following: $a \approx 1 + x\sqrt{dt}$ $V_1 = V(aS, t + dt ) $ The textbook claims you can (using Taylor's Theorem), expand the bottom-most equation like this: $V_1 \approx V + ...
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241 views

Rewrite trigonometric expression to be be numerically “stable”

Is it possible to write the following function: $$ f(x) = \begin{cases} \frac{x-\sin x}{1- \cos x}& x\neq 0\\ 0 & x=0 \end{cases} $$ as a composition of elementary functions (including ...
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2answers
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Find a formula for the binomial coefficients of the Macluarin series for $\frac{1}{(1+x)^{1/2}}$

The Maclaurin series for $\frac{1}{(1+x)^{1/2}}$ is \begin{equation*} 1-\frac{x}{2}+\frac{3x^2}{8}-\frac{5x^3}{16}+\frac{35x^4}{128}...~. \end{equation*} I can't figure much out other than it ...
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How to find the largest positive real value r for Maclauren polynomials where error is set

I need some tips on answering this question: The function is $\sin^2 (r)$ Using Taylor's Theorem, determine the largest positive real value r for which we can guarantee that the Maclauren ...
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2answers
46 views

Taylor expansion of a complex function

Trying to find Taylor series of $$\frac{z^2}{(1+z)^2}$$ I write it in the form $1- \frac{2}{1+z} + \frac{1}{(1+z)^2}$ and I can find Taylor expansion for each factor, is there another method without ...
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2answers
40 views

Taylor series of $1\over z^2$

How to find the Taylor series of $1\over z^2$ near $2$ ( in the power of $z-2$) I have tried to write it in the form: $1\over ((z-2)^2+4z-4)$ But I reached nothing, any help please
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Taylor series in complex analysis [closed]

I am working on finding the Taylor series of $$\frac1{az+b}$$ in powers of $z.$ How to start with it Any help in details...
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1answer
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Taylor series expansion of arctan(x) around the point 0

This is actually a technical question about why I'm getting different results when trying to do the same thing using Maxima and WolframAlpha. When I enter expand arctan(x) in WolframAlpha I ...
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30 views

Theoretical Question regarding Taylor Expansion

I received the following question during a Calculus $2.0$ course in my university. I am not a native speaker, so please excuse my English. The question is as follows: Let $f$ be a function with ...
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45 views

Laurent series for $\frac{2}{(z)(z-1)(z-2)}$

! So I think I am getting the hang of Laurent Series, but having a bit of trouble with one of the fractions for part a). So I split this up in to partial fractions: $\frac{1}{z} - ...
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1answer
39 views

Explanation of taylor series

I understand that for a Taylor series of a function $f(x)$, centered around the point a, the general expression can be written as: $$ \begin{align} &f(x) \\ &= f(a) + f'(a) (x-a) + ...
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1answer
32 views

Find a power series that will converge to F(x)

The question is: Find a power series that will converge to $F(x) = \int_0^x\sin(t^2)\;dt$. I don't really have any idea how to solve this, but I know that I need to create the Maclaurin expansion so ...
3
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1answer
32 views

Upper bound for truncated taylor series

A paper claimed the following but I can't figure out why it's true: For all $1/2> \delta > 0$, $k\le n^{1/2-\delta}$, and $j\le k-1$ where $n$, $j$, and $k$ are positive integers, the following ...