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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3
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1answer
66 views

Determine the sum $\sum_{n=0}^{\infty}{(-1)^n}\frac{2^{2n-1}}{(2n)!}$

I need to calculate the sum $\sum_{n=0}^{\infty}{(-1)^n}\frac{2^{2n-1}}{(2n)!}$ . It seems very "similar" to Taylor expansion of functions arcsin(x) and its derivative for x = -2. It is known: ...
0
votes
1answer
29 views

When is it appropriate to neglect all terms after the first non-zero term of a Taylor expansion series?

Suppose I am interested in the Taylor expansion series of a Cosine function at the neighbourhood of a=0. In computing the series from n=0 to n = infinity, when would it be appropriate to neglect all ...
0
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1answer
21 views

Cropping off the Taylor Series

We know that the Taylor series is for expansion of any function, but for digitization we need to crop off some parts? How can we determine upto which derivative should we consider.. I am mainly ...
0
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0answers
18 views

Using Taylor series of function f1(x) in Taylor series of function f2(x) defined in open discs D1 and D2 when D2 lies inside D1

We have two open discs, D1 and D2, whose centres are C1 and C2 respectively. The Taylor series of function f1(x) is defined in open disc D1 while the Taylor series of function f2(x) is defined in open ...
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5answers
67 views

$\lim_{x \to 0} \cfrac{e^{2x} - \ln(1-x) - \sin(x)}{\cos(x)-1}$ using Taylor Expansions

As a preface- a very similar question is here: Using Taylor expansion to find $\lim_{x \rightarrow 0} \frac{\exp(2x)-\ln(1-x)-\sin(x)}{\cos(x)-1}$ But, my actual question is, when we substitute the ...
0
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1answer
25 views

error term for double integral approximation by midpoint rule

I found following statement in the book that I'm reading: Using Taylor series expansions it is easy to prove that: $$ \left|h^2\cdot ...
2
votes
2answers
78 views

Expand the Taylor series for the following mind-boggling expression at $x = 0$

Mind-boggling expression is: $$f(x) = \frac{x^{15}}{(1-x)(1-x^2)(1-x^3)(1-x^4)(1-x^5)}$$ I started by using the quotient rule and expanding the denominator terms in the hopes of finding some pattern ...
3
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1answer
62 views

How to solve this limit using laurent series?

$$\lim_{x\to\infty}\left(\left(\frac{x^2+5}{x+5}\right)^{3.7}+\left(\frac{x^3+5}{x+5}\right)^{1.6}\right)^{20/37}-\left(\left(x-5\right)^{3.7}+(x^2-5x+25)^{1.6}\right)^{20/37}=60$$ It is possible to ...
0
votes
1answer
31 views

Proving that a function is increasing

I have this problem Suppose that $f: \mathbb{R} \to \mathbb{R}$ is a function, such that its Taylor series convergers to function $f$ everywhere. For every derivative of the function $f$ we have that ...
0
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1answer
28 views

Computing first three non-zero terms of a Taylor series

I have a function $F(t)=\int_0^t \sqrt{1-x^8} dx.$ I have to find the first three non-zero terms of a Taylor series of $F$ around the point $a=0.$ Since I want the Taylor series I started with the ...
0
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1answer
33 views

Does an inequality between first-order Taylor approximations imply the same for the functions?

Assume that $f$ and $h$ are functions from $\mathbf{R}^n$ to $\mathbf{R}^1$ and continuously differentiable. Also assume that $f(z)=h(z)$ at some point $z \in \mathbf{R}^n$. Could we then show that ...
1
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2answers
36 views

Extending a bounded holomorphic function past its boundary

Suppose I have a bounded holomorphic function on the unit disc, centred at the origin. Can I always extend this beyond the origin to say a disc of radius $1 + \epsilon$ for some $\epsilon > 0$? My ...
0
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1answer
18 views

Taylor expansion of $f$ in stability analysis of 2-step Adams-Bashforth method

Given the two-step Adams-Bashforth method $$ u_{n+1} = u_n + \tfrac{h}{2}(3f_n - f_{n-1}) $$ find its order. Some notation: $t_n = t_0 + nh$ is the $n$-th node and $y_n = y(t_n)$; $f_n$ ...
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0answers
23 views

Does small perturbation in the denominator explode the expectation of a ratio of two random variables?

The puzzling thing I am facing is Suppose we have two random variables $X$ and $R$ such that $E(X^{-1}R)=1$. Now let $\tilde{X}=X+\mathcal{E}$ where $\mathcal{E}=X\epsilon$ and $\epsilon \sim ...
0
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1answer
17 views

Index confusion in Tu's treatment of Taylor's Theorem with remainder in “An Introduction to Manifolds”

In Tu's book, specifically the section on "Taylor's Theorem with remainder", there appears to be a changing of the meaning of some subscripts which isn't noted. The theorem states that if $f$ is a ...
2
votes
1answer
37 views

Taylor expansion at discontinuous point

a) Find the Maclaurin expansion of the following function: $$f(x)=\int\limits_0^x \frac{1-e^{-t^3}}{t^2} \mathrm{d}t$$ end b) evaluate the $ \displaystyle \lim_{x \to 0^{+}} f^{(29)}\, (x) $ The ...
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1answer
36 views

approximation of $\pi$ by $\arctan$

Determinate the order n of the Maclaurin polynomial for $f(x)=4tan^{-1}x$ so that the remainader term $|R_{n}(1)|<0.000005$. Here $R_{n}(1)=\frac{f^{(n+1)}(c)}{(n+1)!}$ for some c between 0 and 1 ...
4
votes
1answer
104 views

Taylor series approximation of function under norm

I am reading this paper. At page number $4$, term $||Au - f||^{2}$ is approximated by taylor series approximation around $u^{k}$. The resulting approximations are $$\|Au - f\|^{2} \approx ...
2
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0answers
32 views

Steady state response approximation of a linear differential equation using Taylor polynomial

After thinking out how to convert a non-homogeneous linear differential equation, with a polynomial input, to a homogeneous linear differential equation in general for this question I started playing ...
4
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0answers
89 views

On applying Whitney's extension theorem to suitable closed sets

Whitney's extension theorem states that if $D \subset \mathbb{R}^n$ is closed and $f: D \to \mathbb{R}$ is $C^k$ in some sense to be specified below, then $f$ can be extended to $\mathbb{R}^n$ so that ...
0
votes
1answer
13 views

Use the remainder estimation theorem to find the maximum value of error?

My answer for part (bi) was $\frac{x}{2}$ - $\frac{x^2}{4}$ My attempt for part b(ii) was to find $\frac{g^{'''}(z)}{3!}$($x^3$) = $\frac{8}{3!(2+2z)^3}$ where z ∈ [0, $\frac{1}{2}$]. To find the ...
0
votes
1answer
22 views

The Maclaurin series and taylors theorem for $\sinh(2x)$

I am currently studying for an exam next week but am struggling to the second part of this question. I have figure out the Maclaurin series for $\sinh(2x)$, however am unsure how to estimate the ...
0
votes
1answer
19 views

Calculate the 3rd order Taylor polynomial about $x=1$?

Calculate the $3$rd order Taylor polynomial about $x=1$ for the function $f:[-3:\infty) \longrightarrow \mathbb{R}$ given by $f(x)=\sqrt{x+3}$. I know that the formula for the Taylor Polynomial ...
0
votes
1answer
32 views

Calculate the Taylor series of $\sin$ around $3$

This should be simple but I'm having trouble with it. So by definition the series looks like $$\sin 3 + \cos (3) (z-3) - \frac{\sin (3) (z-3)^2}{2!} - \frac{\cos (3) (z-3)^3}{3!}+...$$ To be able ...
2
votes
1answer
72 views

Approximation for probability of at least $t$ events

I'm reading through a paper, and they are discussing the approximate probability that $t+1$ out of $t^b$ events occur, where $b$ is a constant, and the probability of each event occurring is ...
0
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0answers
26 views

How do I expand this function?

Can anyone help me to expand this function: $$ f(\theta) = R \epsilon_0 \left(\frac{\ln{ (1+L/d \tan{( \theta )})}}{\tan{(\theta)}}\right)$$ I want to expand it for small $\theta$ (I guess around ...
0
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0answers
31 views

Determine the 19-th, 20-th and 21-st order Maclaurin polynomial?

The answers tell me that the solution is: Firstly, I don't understand the notation "x$\to$". What does this mean? This is my attempt to find the maclaurin polynomial for x(1-cos2$x^3$): ...
7
votes
2answers
113 views

Taylor series not converging, other example than $\exp(-1/x^2)$?

The usual example for non-converging Taylor series is $g(x) = \exp(-1/x^2) \; \forall x \neq 0, g(0) = 0$: the Taylor series around $x=0$ is zero, but $g$ isn't zero for any $x \neq 0$. What's not so ...
0
votes
1answer
61 views

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 ...
0
votes
2answers
23 views

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 ...
1
vote
1answer
71 views

Coefficient of operator and how to do it

This question stems from this $$ \frac{1}{x+z}- \frac{1}{x} = \sum_{k=0}^\infty \frac{z^k}{k!}\frac{d^k}{dx^k}[\frac{1}{x}] $$ Now, i need to find the Bell Polynomial of $\frac{1}{x}$, $$ ...
3
votes
3answers
60 views

How to expand $\sqrt{x^6+1}$ using Maclaurin's series

The expansion would be $\sum_{n=0}^\infty$$\frac{1}{2}\choose n $$x^{6n}$ How to evaluate binomial coefficient with rational numbers? If $\frac{1}{2}\choose n $=$2n\choose n $$\times ...
0
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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 ...
1
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1answer
48 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. ...
0
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2answers
39 views

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 ...
1
vote
2answers
43 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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0answers
13 views

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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0answers
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 ...
2
votes
1answer
35 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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0answers
11 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 ...
6
votes
0answers
74 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 ...
1
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1answer
44 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,
3
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1answer
59 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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0answers
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) - ...
8
votes
3answers
2k views

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 ...
0
votes
2answers
36 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}+...\\ ...
1
vote
2answers
46 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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0answers
19 views

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
1
vote
2answers
36 views

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, ...
5
votes
1answer
64 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 ...