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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Convergence radius of $\sum_{n=1}^\infty (-1)^nn^nx^{n^2} $

I need to find the convergence radius of the series $$\sum_{n=1}^\infty (-1)^nn^nx^{n^2} $$ I have tried using the ratio test $$ \lim\limits_{n \to \infty} \lvert \frac{a_{n+1}}{a_{n}}\rvert $$ and i ...
2
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0answers
22 views

terms of taylor expansions of multiple variables at the origin

By the fundamental theorem of symmetric polynomials, $X_1,X_2,\cdots,X_n$ are polynomials of $ e_1,\cdots,e_n$ and $$ \mathbb{Z}[ e_1,\cdots,e_n]=\mathbb{Z}[X_1,X_2,\cdots,X_n]. $$ We define a ...
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0answers
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How can we show that $\sum\limits_{n=1}^\infty |c_n| < \infty$ when $c_n$ is taylor series coefficients of $f(z): \mathbb{C} \rightarrow \mathbb{C}$?

I'm trying to understand a solution to old exam question. And i have trouble understanding and verifying (for myself) one of the steps. if $f(z): \mathbb{C} \rightarrow \mathbb{C}$ is holomorphic on ...
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1answer
35 views

Evaluate the following limit using Taylor series.

What is the limit, when $x\to0$, of $$\frac{4\tan x - 4x -\frac{4}{3}x^3}{x^5}?$$ I'm not sure how to expand this using the Taylor series.
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1answer
51 views

An error bound for a finite difference approximation to the second derivative

I want to show that $$|\delta_{h,r}f(x)- f''(x)| \leq \frac{11}{12} h^2 \|f^{(4)}\|_{\infty}$$ where $$\delta_{h,r}f(x)=\frac{1}{h^2} (2f(x)-5f(x+h)+4f(x+2h)-f(x+3h))$$ I applied the Taylor expanson ...
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1answer
30 views

Find limit using Maclaurin series (remember the importance of big O notation)

I have a problem that sounds like this: Find the limit $$\lim_{x\rightarrow 0} \frac{14\tan(6x)-84x}{6x^3}$$ using Maclaurin series, and don't forget the importance of big O notation. I have tried ...
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1answer
33 views

How to find the best value of K ? (Big-O)

Can anyone help me with finding out the best value of K in the following assertion as x goes to 0: $$cos(x)-1+x^2/2=O (x^k)$$ Thanks in advance! Ali
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2answers
98 views

Convert Power Series to function

I tried to solve the attached Power Series, however I can't get to the right answer. I wrote down the correct answer at the top-right of the page. Appreciate your help!
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1answer
47 views

How to expand $x/\sin x$?

Please help me, I have no idea how to solve this. I know the expansion of $\sin x$ but am not sure if it will apply for $1/\sin x$.
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2answers
92 views

How to fastest approximate definite integrals

I know that a definite integral is a limit of Riemann sums. So if one wanted to estimate a definite integral (because one might not be able to find an antiderivative), then one can just take enough ...
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1answer
56 views

Number of terms of $\sin(x)$ required for maximum error of less than $10^{-7}$

Neglecting round-off error, how many terms of the Maclaurin series for $\sin x$ are required to obtain a maximum error of less than $10^{-7}$ in the range $[0,\pi]$? This is part (b); in part (a), I ...
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1answer
35 views

What is the expansion?

I encounter the following formula in some textbook. However, I can not understand what the expansion in this formula is. Is there anyone giving some tips? $$ \begin{align} \alpha &= -\frac{iU}{2 ...
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1answer
39 views

Help me show that there exists an $x\in (\pi/2,\pi)$ such that $\sin x=\frac{x}{2}$

Question Show that there exists an $x\in (\pi/2,\pi)$ such that $\sin x=\frac{x}{2}$ My Solution The way I did it is: $$\sin x=\frac{x}{2}$$ $$x=2\sin x$$ Using Taylor Series ...
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2answers
67 views

How is Taylor expansion a generalization of linear approximation? [closed]

The concept of derivative is related to linear approximation of a function: $$f(x)\approx f(a)+f'(a)(x-a)$$ I was told that this linear approximation is generalized by the Taylor series. What does ...
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0answers
22 views

$\lim_{x\rightarrow x_0 }\frac{f^{(k)}(x_0)x^k + f^{(k+1)}(c)(x-x_0)^{k+1}}{g^{(k)}(x_0)x^k + g^{(k+1)}(c)(x-x_0)^{k+1}}$

Doing a division $\frac{f(x)}{g(x)}$ of two Taylor polynomials of functions $f$ and $g$ of the form $$f(x) = T_{n,x_0}f(x)+R_{n,x_0}f(x) = \frac{f^{(k)}(x_0)}{k!}x^k + ...
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3answers
65 views

Taylor series of arctan(x) (Spivak)

At p. 388 of Calculus, Spivak gives a formula: $$\frac{1}{1+t^2} = 1 - t^2 + t^4 - ... + (-1)^nt^{2n} + \frac{(-1)^{n+1}t^{2n+2}}{1+t^2}$$ Which can be integrated to find $\arctan(x)$. I don't ...
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1answer
17 views

Big-O division in $\frac{f(x)}{g(x)}$ of Taylor polynomials

Here p.4 bottom is a proof that proves a property of $\frac{f(x)}{g(x)}$, the quotient of two Taylor polynomials of f and g. For two Taylor polynomials of $f(x) = \frac{f^{(k)}(x_0)x^k}{k!}+o(x^k)$ ...
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1answer
35 views
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33 views

Question on the deduction of the integral form of remainder of Taylor series

[Theorem deduction] First we have a generalized form of integration by parts $$ \begin{align} \int u(x)v^{(n+1)}(x)dx=\sum_{i=0}^n(-1)^iu^{(i)}v^{(n-i)}+(-1)^{n+1}\int u^{(n+1)}(x)v(x)dx &&(G) ...
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0answers
23 views

Domain specification of derivative extension.

Given the definition of Taylor expansion: $$f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x-a)^n$$ We can find the $m$'th derivative of $f(x)$ quite easily: $$\frac{d^m}{dx^m} f(x) = ...
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2answers
41 views

A particular Taylor Expansion

As we know, the Taylor Expansion we usually see is \begin{equation} e^W = \sum_{n=0}^{\infty}\frac{W^n}{n!} \end{equation} but today I see another equation: \begin{equation} e^{-W}+We^{-W} = ...
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0answers
24 views

How to write the symmetric Hessian matrix for a log function?

Say f(x,y,z) = $y*ln(cos(z)+x^2)$ How would I write this as a Hessian matrix? Would this be the right step I need to take in order to calculate the second-order Taylor polynomial for the function?
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1answer
117 views

Estimating error in binomial series

I am having trouble with estimating the error in this exercise: Estimate the value of the following integral with error less than 0.0001 $$ \int_{0}^{0.2}\sqrt{1+x^3} \, dx $$ So here im using ...
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2answers
24 views

Taylor polynomial of order $n$ for a polynomial of degree $n$

I noticed that the Taylor polynomial of order $n$ for a polynomial function of degree $n$ is identical to the function. I tried to understand the reason but couldn't really figure it out. Any input on ...
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3answers
49 views

How do you write the taylor series for (arctan(x))^2

I know that the taylor series of $\arctan{(x)}$ is $x - \frac{x^3}3 + \frac{x^5}5 + \dotsc$ In order to square it I would have to multiply it by itself. $(x - \frac{x^3}3 + \frac{x^5}5 + \dotsc) ...
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1answer
25 views

Taylor expansion with geometric series [closed]

Expand $1/x $ a taylor series around $x=1$. Use $\frac {1}{1-(1-x)}$ and used the geometric series. I know $\sum (1-x)^n = \frac {1}{1-(1-x)}$, $-1 <x -1<1$
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1answer
102 views

Finding an upper bound on the Taylor remainder

Any help with the proof I have posted for the following question is greatly appreciated; I have listed my particular issue at the end. Thank you! Let $p \geq 0$. For $n=1,2, \dots,$ find $P_n(x)$ ...
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0answers
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If $0< 2\varepsilon < \sigma^2 < 1$ then $\prod\limits_{i = 1}^n (1 + \varepsilon + \sigma \xi _i )$ converges almost surely to $0$

I posted this question a few days ago and there were some errors in my post. I have fixed them and it should be all right now. Hope someone can help with my confusion. Let $(X_n)_{n\ge0}$ denote an ...
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1answer
41 views

Why can't I get good approximation when choosing values away from point of expansion? (Taylor series)

I was in the middle of doing a computing project assigned to me when I came across the question. $\operatorname{P}_N(x)$ is the taylor polynomial for $f(x)=\ln(x)$ expanded around pouint $x_0=1$ ...
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0answers
32 views

Find the cubic Maclaurin polynomial for $\sinh x$ and estimate the remainder on the interval $|x| \le 1 $

I'm working through the above problem, and am having trouble deciphering what it means by "estimate the remainder on the interval $|x| \le 1$". I found the Maclaurin polynomial to be $$P_3(x) = x + ...
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1answer
32 views

Taylor's Theorem for 2D function

I am doing a numerical analysis course and we are looking at numerical methods for solving initial value problems. For example: methods such as $y_{n+1} = y_n+\frac{h}{2}(f(t_n,y_n) + f(t_{n+1}, ...
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1answer
39 views

Proofs for Taylors theorem and other forms

Let $f \in C^k[a,b]$.Show that for $x,x_0 \in [a,b]$, $$f(x)=\sum\limits_{j=0}^\mathbb{k-1}{{1\over j!}f^{(j)}(x_0)(x-x_0)^j}+{1\over k!}{\int_{x_0}^x f^{(k)}(t)(x-t)^k \,dt}$$ and after this use this ...
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0answers
177 views

Proof that $\oint_r d(x,N + n) < 0 $?

Let $f(x)$ be a real-entire function such that for all $x>0$ we have $f(x) > 0$, $f'(x) > 0$ , $f '' (x) > 0$. And also $0 < D^M f(0) < D^{M-1} f(0)$. Let $0<T<1$ and $n$ a ...
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1answer
36 views

how to find taylor serie for 1/z with |z| > 0?

I have the following and I need to give the Laurent development for |z| > 0. The Laurent development in this form : and to give few a(n) coefficients How can it be done? normally we use the ...
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2answers
26 views

Taylor series, identify radius of convergence

I have the following function : I need to find it's radius of convergence with z0 = 0. The function is analytic everywhere except where 1 + sin(iz) = 0 (to my ...
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1answer
59 views

Laurent-series expansion of $\frac{1}{(e^z-1)^2}$ about $z=0$

I am studying for exams in complex analysis and taking a look at past papers. This comes up often or an integral of the given function along a certain curve, which is actually the same thing since the ...
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4answers
56 views

Find the Taylor series about $x = 1$ for $f(x) = \dfrac{1}{(x − 2)^2}$ . [closed]

Find the Taylor series about $x = 1$ for $f(x) = \dfrac{1}{(x − 2)^2}$ . Express your answer in sigma notation, simplified as much as possible. This is a practice question that I am having trouble ...
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1answer
18 views

sin(x+y^2) taylor expansion little oh error term degree >3

I am trying to understand example 3.4.5 in John and Barbara Hubbard's second edition of Vector Calculus, Linear Algebra, and Differential Forms. It provides the taylor expansion of $sin(x+y^2)$ by ...
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1answer
73 views

Taylor series expansion?

How to find the Taylor series expansion of $$(1+x)^{1/x}$$ I tried with the Taylor series but unable to solve it. Help me out. Hints or anything that sort will be helpful.
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2answers
61 views

Taylor expansion of logarithm function.

Expand $f(x) = \log(1 + x)$ around $x = 0$ to all orders. More precisely, find $a_n$ such that for any positive integer $N$, we have$$f(x) = \left(\sum_{n=0}^{N-1} a_nx^n\right) + E_N(x) \text{ for ...
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1answer
70 views

How to compute $\lim\limits_{x\to 0}\dfrac{e^{f(x)}-e^x}{2x-\sin\left( f(2x) \right)}$

Let $f:\mathbb{R}\longrightarrow \mathbb{R}$ a function such that : $f(x)=x-x^3+o(x^3).$ Compute $$ \lim\limits_{x\to 0}\dfrac{e^{f(x)}-e^x}{2x-\sin\left( f(2x) \right)}$$ My thoughts: ...
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1answer
84 views

Behaviour of $f(x)=e-\left(1+\frac{1}{x} \right)^{x}$ when $x\to+\infty$

This is from an MCQ contest. For all $x\geq 1$ let $f(x)=e-\left(1+\dfrac{1}{x} \right)^{x}$ then we have : $f(x)\mathrel{\underset{_+\infty}{\sim}}\dfrac{e}{x}$ and $f$ is integrable on ...
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0answers
37 views

Integrability of $f(x)=\left(1+\frac{1}{x} \right)^{1+\frac{1}{x}}-a-\frac{b}{x}$

This is from an MCQ contest. For all $x\geq 1$ let $$f(x)=\left(1+\dfrac{1}{x} \right)^{1+\dfrac{1}{x}}-a-\dfrac{b}{x}$$ note that ...
3
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1answer
38 views

Does Cauchy's estimate imply analyticity?

Komatsu says here (Proc. Japan Acad. Volume 36, Number 3 (1960), 90-93) that a smooth function which satisfies Cauchy's estimate is analytic. How does one prove this? Surely, if Cauchy's estimates ...
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1answer
118 views

Taylor expansion for $\arcsin^2{x}$

I stumbled upon this particular expansion that was included in this post. $$ \displaystyle \arcsin^{2}(x) = \frac{1}{2} \sum_{n=1}^{\infty} \frac{1}{n^{2} \binom{2n}{n}} (2x)^{2n}$$ This caught ...
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0answers
123 views

Finding closed-form approximations of the solutions of $f(x,y)=0$

Consider $$f(x,y)=\sum_{i=1}^n\dfrac{\sin(\omega_ix)}{\sin(\omega_iy)}r_i$$ where $n,\omega_i,r_i>0$ are known parameters. Restrict to domains where $ \sin(\omega_iy)\neq 0$, and by symmetry ...
2
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1answer
38 views

When is $a(z) = b(c(z)) $?

Let $a(z)$ be a given transcendental entire function. When is $a(z)=b(c(z))$ where $b,c$ are also transcendental entire functions ? How to find such $b,c$ ? In particular when $a$ is given by a ...
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0answers
25 views

Central Difference taylor approximation

We are asked to show that We have so far managed to show the first two equalities using finite difference approximations but the last one still eludes us. Any hints?
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25 views

Generalised Taylor series to fractional order derivatives and special functions

A year ago or so I read this papar which was wonderfully illuminating link. For example the author seduces the reader with wonderfully compact representations like that of the bessel $J_v(z)$ function ...
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1answer
57 views

Looking for Taylor series expansion of $\ln(x)$

We know that the expansion of $$\sin(x) $$ is $$x/1!-x^3/3!\cdots$$ Without using Wolfram alpha, please help me find the expansion of $\ln(x)$. I have my way of doing it, but am checking myself with ...