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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What is the 2nd order taylor polynomial of f(x,y)?

I'm just computing the 2nd order taylor polynomial for $f(x,y) = tan(x + 3y + \frac{\pi}{4})$ centered at (3,-1) and wondering if I have done this correctly or if anyone has any suggestions on how I ...
3
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2answers
50 views

Finding terms of a Taylor series where $f(x)$ is a function with a power

I've been stuck with this Taylor series problem for a while now. We have that $$ f(x) = (1 + x^2)^{-2/3} $$ and it's centered at $0$. So what I thought of doing was the $$ \frac{f^{n}(a)(x - ...
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1answer
36 views

Is this series G(1/n) convergent or divergent given G(x)?

Suppose $G(x)=\int_0^x\sin{\left(e^s-1\right)}ds$ Does the series $\sum_{n=1}^{\infty}G(\frac{1}{n})$ converge or diverge? I'm not sure how to go about solving this; however in our notes it says ...
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3answers
24 views

Question on a Taylor Polynomial

We are asked to generate the taylor polynomial $P(x)$ for $$ f(x) = \frac{e^{{(x-1)}^2}-1}{(x-1)^{2}} $$ about $x=1$ Using substitution into the known taylor polynomial of $e^{x}$ and further ...
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2answers
35 views

Taylor polynom, residual for symmetric values

When creating the taylor polynom for a $C^3$-function around a certain point i get the formula $f(z+h)=f(h)+hf'(z)+\frac{h^2}{2}f''(z) + \frac{h^3}{6}f'''(z) + R$ Now lets say I create the polynom ...
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3answers
194 views

Find complicated Taylor Series

According to some software, the power series of the expression, $$\frac{1}{2} \sqrt{-1+\sqrt{1+8 x}}$$ around $x=0$ is $$\sqrt{x}-x^{3/2}+\mathcal{O}(x^{5/2}).$$ When I try to do it I find that I ...
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3answers
38 views

standard Taylor series using substitution

Find Taylor series using substitution about $0$ for $f(x)=\frac{125}{(5+4x)^3}$ by writing $\frac{125}{(5+4x)^3}=\frac{1}{(1+\frac{4}{5}x)^3}$? Determine a range of validity for this series.
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1answer
74 views

Finding an asymptotic expansion for a transcedental equation

I am new around here and was hoping you will be able to help me with the following. I have the equation: $x^3 - 3x^2 +(3-\epsilon ) x + \epsilon = sin(\frac{\pi}{2} x +\frac{\pi \epsilon}{2} ) $ and ...
2
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6answers
252 views

Definition of matrix exponential

Is there an alternative definition of a matrix exponential so I can use it to prove that $$e^{A}=\sum_{m=0}^{\infty} \frac{1}{m!}(A)^m \;?$$ Thanks a lot in advance!
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5answers
105 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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1answer
16 views

Taylor Expansion and Log transformation (Time Series)

From: Time Series Analysis with Applications in R by Jonathan D. Cryer and Kung-Sik Chan. Here is the Taylor expansion: $\log Y_t = \sum_{n = 1}^{\infty} (-1)^{n+1} \frac{(Y_t - 1)^n}{n} $. How ...
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2answers
36 views

what is the Maclaurin Series of this function?

Can anyone explain to me how to find the Maclaurin series of: $$f(x)=(x^2+1)e^{\frac{-x^2}{4}}$$ and why does it converge for every x? thanks,
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1answer
39 views

Dot product of taylor series $\sqrt{1+x}$

I have to prove that $$ \sum_{k=1}^n \alpha_k \cdot \alpha_{n-k+1} = 0, $$ where $n>2$ and $\alpha_k$ is the k-th member in taylor series of $\sqrt{1+x}$. Namely, $$ \alpha_k = ...
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1answer
40 views

Problem with Taylor (asymptotic) expansion of hyperbolic functions at infinity

(Note: I chose a general title, because I believe this discussion will be applicable to all other hyperbolic functions having an asymptote at infinity, but I will specifically be focusing on ...
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1answer
71 views

Taylor Series Expansion for $\tan x$

I'm trying to determine the Taylor series expansion for $\tan x$: I know that the $n$th derivative of the expansion must be the same as the $n$th derivative of the function. Please help, I have no ...
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1answer
180 views

Limits of the solutions to $x\sin x = 1$

Let $x_n$ be the sequence of increasing solutions to $x\sin{x} = 1$. Define $$a = \lim_{n \to \infty} n(x_{2n+1} - 2\pi n) $$ and $$b = \lim_{n \to \infty} n^3 \left( x_{2n+1} - 2\pi n - \frac{a}{n} ...
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1answer
13 views

Showing a function is identically zero using taylors theorem

Suppose we have a function $x(t)$ that is analytic everywhere, $x(t)\geq 0$, and $x(0)=0$. Suppose further that we know $x(t)$ is locally zero. ie. $x(t)=0$ for some small $t>0$. Is there a way ...
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1answer
41 views

Proving $\frac{\cos(t \arctan(\sqrt{x}))}{(1+x)^{t/2}}= \sum_{k \ge 0}\frac{\Gamma(t+2k)\Gamma(k+1)}{\Gamma(t)\Gamma(2k+1)}\frac{(-x)^k}{k!}$

$$\frac{\cos(t \arctan(\sqrt{x}))}{(1+x)^{t/2}}= \sum_{k \ge 0}\frac{\Gamma(t+2k)\Gamma(k+1)}{\Gamma(t)\Gamma(2k+1)}\frac{(-x)^k}{k!}$$ This is particularly nice, and apparently can be proved with the ...
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1answer
88 views

How would I integrate $e^{e^x}$?

Is there a way to integrate: $e^{e^x}$ without using a Taylor or McLaurin Series expansion?
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0answers
13 views

How can I show that $u=e^{\sigma\sqrt{\Delta t}}$ in the binomial option pricing model

Given that $e^{r\Delta t}(u+d)-ud-e^{2r\Delta t} = \sigma^2\Delta t$ I would like to show that $u=e^{\sigma\sqrt{\Delta t}}$ I know I must somehow use Taylor's approximation $e^x = 1 + x + ...
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1answer
39 views

Taylor series of a rational function

I am facing some complicated integral, which part of it is $$\frac{z^{M-1}}{(1+(\eta z)^n)^p}$$ I think if I find the taylor series of this part the integral might be solved. So, can someone help me ...
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1answer
33 views

Finding $\frac{\partial ^8 f}{\partial x^4\partial y^4}$

Given the function $f(x,y)=\frac{1}{1-xy}$ find the value of$\frac{\partial ^8 f}{\partial x^4\partial y^4}(0,0)$. First I developed the function into a taylor series using geometric series ...
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1answer
86 views

Taylor Expansion of $ \frac{1}{x} $ about x = 0

I'm confused with the following problem: The expression $ \frac{1}{x} $ is clearly not defined at x = 0. However, I read that it could be expressed as a series using the idea $ ...
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4answers
84 views

Taylor Series of $ \frac{1}{1-x^2} $ about x=2

I am trying to form a taylor series of the following: $ \frac{1}{1-x^2} $ about $x=2$ I tried factoring the equation such that it becomes the following: $ \frac{1}{{(1+x)}{(1-x)}} $ I tried to ...
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1answer
48 views

Does Taylor's theorem apply here?

Let $U\subset \mathbb{R}^n$ be open and $f:U\to \mathbb{R}^n$ with $x\in U$ and $\xi$ sufficiently small. Suppose that the following hold: $f(x+\xi)=\sum_{\alpha=0}^k ...
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3answers
171 views

Find a power series representation for the function $f(x)=\frac{(x-1)^2}{(3-x)^2}$

I tried to separate it and found the sum of $$\frac{1}{(1-x/3)^2}$$ but then I got stuck with having to multiply my sum with $(x-1)^2$ . I tried looking online but there's close to nothing about ...
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1answer
84 views

How to find the series $\sum_{n=1}^{\infty}\frac{[(n-1)!]^2}{(2n)!}(2x)^{2n},$

Find this sum $$\sum_{n=1}^{\infty}\dfrac{[(n-1)!]^2}{(2n)!}(2x)^{2n}.\qquad (-1\le x\le 1)$$ My idea: let $$f(x)=\sum_{n=1}^{\infty}\dfrac{[(n-1)!]^2}{(2n)!}(2x)^{2n},$$ then we have ...
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3answers
61 views

Why Taylor series does not converge for all x in the domain of the function

Example: $$ f(x)=\frac{1}{1+x} \qquad x\neq-1 $$ $$ f(x)=1-x+x^2-x^3+x^4-x^5+\;... \qquad |x| < 1 $$ Why Taylor series does not converge for all x in the domain of the function?
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1answer
39 views

Prove inequality using taylor series

Let $0\leq p \leq 1$ and $\phi(t)=t\log \frac{t}{p} + (1-t)\log \frac{1-t}{1-p}$. Prove $\phi(t)\geq 2(t-p)^2$ for $t\in[0,1]$. Here's how I started. $$\phi'(t) = -\log ...
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0answers
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Taylor’s theorem with the Lagrange form of the remainder Expansion

Write down the Taylor expansions of $f(0) and f(2)$ using Taylor Theorem with the Lagrange Form of the remainder. Here is the formula. $f(a+h)=f(a)+hf'(a)+ (1/2) h^2f''(a+θh)$ My confusion is ...
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0answers
58 views

Use Taylor Theorem Special Form to Prove

Suppose $f: \mathbb{R} \rightarrow \mathbb{R} $ is such that both $f'$ and $f''$ exist for all $x \in \mathbb{R}$, Suppose that on [0,2] the inequalities $|f(x)|\leq 1$ and $|f''(x)|\leq1$ hold. ...
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0answers
74 views

Use Taylor Theorem and Taylor Expansion to Prove

Let $g: R\rightarrow R$ be a twice differentiable function satisfying $g(0)=1, g'(0)=0$ and $ g''(x)-g(x)=0$, for all $x$ in R Fix $x$ in R. Show that there exists $M>0$ such that for all natural ...
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7answers
770 views

How are the Taylor Series derived?

I know the Taylor Series are infinite sums that represent some functions like $\sin(x)$. But it has always made me wonder how they were derived? How is something like ...
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1answer
36 views

Alternating series error bound

The taylor series for $ln(x)$, centered at $x=1$, is $$\sum_{n=1}^{\infty}(-1)^{n+1}\frac{(x-1)^n}{n} $$ Let $f$ be the function given by the sum of the first three nonzero terms of this series. The ...
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0answers
32 views

Finding f'(0) in a taylor series

While doing questions involving taylor series, I accidentally chanced upon an unorthodox, if more difficult way of solving for $f^{(n)}(0)$ of a given taylor series. I am wondering why it works, if ...
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2answers
29 views

Generating a general term for a taylor polynomial

Let $f$ be the function given by $$ f\left(x\right) = \sin\left(5x +\frac{ \pi }{4}\right)$$ Let $P\left(x\right)$ represent Taylor polynomial of $f$ centred at $x =0$. Generate the general term for ...
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0answers
79 views

Taylor series expansion for $e^{\sin{x}}$ [duplicate]

Given the function $$f(x)= e^{\sin{x}}$$ I have to write it without using the exponential or sine function. I came to this point $$f(x) = \sum_{k=0}^{\infty} \frac{\sin^k{x}}{k!}$$ How can I get ...
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1answer
46 views

Generating Functions Interpretation - Expanding around other points?

Generating functions are incredibly useful for solving all kinds of combinatorial problems. Whenever they are used, though, the generating function is always expanded around $x=0$ to obtain the ...
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2answers
40 views

Taylor series expansion and the radius of convergence

Hello I have some problems concerning Taylor series. Given the function $$f(x)=e^{\sin{x}} $$ I concluded that the Taylor series expansion would be $$f(x) = ...
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1answer
17 views

Reference of the expansion of square root polynomials

What is the reference of the formulation given below by Robert israel, please inform me.. Given an even-degree polynomial $$P(x) = a_{2n} x^{2n} + a_{2n-1} x^{2n-1} + \ldots + a_0 = x^{2n} (a_{2n} + ...
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1answer
130 views

Taylor Series of $\frac{1}{1-\cos x}$

The problem is, as the title suggests, to find the Power Series Expansion of $\frac{1}{1- \cos x}$ around $x=c$. What I've tried: Direct Computation: Derivatives get very ugly quickly, and don't ...
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2answers
58 views

Using Lagrange form of the remainder with cosh

I am trying to find "$\cosh 4$ using the sixth partial sum ($n=5$) of the Maclaurin series" for the function. I am also trying to use "the Lagrange form of the remainder to estimate the number of ...
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0answers
14 views

How can i Taylor expand to get a difference approximation formula using

How can i Taylor expand to get a difference approximation formula using $y'''(0)=ay(-h)+by(0)+cy(h)+dy(2h)+O(h^p)$ where $O(h^p)$ needs to be as high as possible? i.e how can i Taylor expand the ...
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1answer
21 views

Lagrange Remainder and Intervals of convergence

(a) Determine the largest interval centered at $c=0$ on which we can be sure that $\lvert \cos(x) -(1-\frac{x^2}{2})\rvert < 10^{-6}$ (b) Let $T_n(x)$ denote the Taylor polynomial of order $n$ for ...
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1answer
40 views

Convergence question and degree of polynomial

I'm currently teaching myself power series and Taylor's theorem for complex analysis and I'm having trouble answering questions of the following form: $1)$ Suppose the power series ...
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0answers
40 views

Why this function is elementary and its pair is not?

Why $$f(x)=\frac{2 \zeta (2)}{\pi ^2}+\frac{6 \zeta(4) x^2}{\pi^4}+\frac{10 \zeta(6) x^4}{\pi^6}+\frac{14 \zeta(8) x^6}{\pi^8}+\cdots$$ is elementary while $$g(x)=\frac{4 \zeta (3)x}{\pi ...
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2answers
48 views

What is series coefficient for $f(x)=\csc^2 x - \frac1{x^2}$?

What is general formula for Maclauren series expansion for $f(x)=\csc^2 x - \frac1{x^2}$ ?
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2answers
28 views

Taylor Expansion of $\frac{x^4}{9+x^3}$ using elementary series

I have exhausted my book of tricks trying to do a series expansion of: $$f(x)=\frac{x^4}{9+x^3}$$ It is trivial to obtain by taking successive derivatives of the function, but I would like to know ...
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1answer
52 views

Taylors theorem application [duplicate]

I posted this question yesterday, and, despite getting answers, I am still confused how to solve it: Use Taylor's theorem to prove that $\displaystyle\lim_{n \to \infty} n ...
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votes
3answers
68 views

Proof using Taylor's theorem

Use Taylor's theorem to prove that $\displaystyle\lim_{n \to \infty} n \ln\left(1+\frac{1}{n}\right)=1$ I don't understand how to apply Taylor's theorem to a limit, especially one with a product of ...