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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Logarithmic Taylor series question [closed]

Consider the transformation of variables, x = $\frac{y+1}{y-1}$ How would you develop log(x) as a Taylor Series in y about zero?
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104 views

How to see $\cos x \leq \exp(-x^2/2)$ on $x \in [0,\pi/2]$?

Can anyone help me with the above inequality? I tried looking at the series expansion and I guess the answer indeed lies there, but I fail to see it. Thanks
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19 views

Derive orthogonal transformation

Let $R$ and $R'$ be two cartesian co-ordinate systems and $\phi=(\phi_1,\phi_2,\phi_3):\mathbb{R}^3\to\mathbb{R}^3$ a map that relates the $\textbf{x}=(x_1,x_2,x_3)$ co-ordinates of $R$ with the $\phi(...
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0answers
29 views

How do we know that a function can be written as a power series?

Most proofs of a Taylor series or a Maclaurin series assume that the function can be written as a power series. If a function can be written as a power series then: $$f(x)=\sum_{n=0}^\infty \frac{f^{(...
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1answer
98 views

Proving analyticity of an integral function over $\mathbb{R}^{n}$

Let $U\subsetneqq\mathbb{R}^{n}$ be open, $\varepsilon>0$ and consider the function $$f_{\varepsilon}(x)=\frac{\pi^{-\frac{n}{2}}}{\varepsilon^{n}}\int_{U}\exp\left\{-\left\|\frac{x-y}{\varepsilon}\...
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3answers
150 views

Solving the second taylor polynomial

So I've found myself in a predicament when trying to implement the second Taylor polynomial. Here is my question: Let $f(x) = \sqrt{x}$, find the second Taylor polynomial $P_2(x)$ for this ...
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1answer
37 views

Why is $\prod_{i=1}^{k-1}\left(1-\frac{i}{2N}\right) \approx 1 - \frac{{k \choose 2}}{2N}$?

I came across this approximation in the book Principles of Population Genetics by Hartl and Clark (page 130). $$\prod_{i=1}^{k-1}\left(1-\frac{i}{2N}\right) \approx 1 - \frac{{k \choose 2}}{2N}$$ ...
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1answer
51 views

Determineing the largest number such that the Laurent series of converges for a trig function.

Question How to determine the largest number $R$ such that the Laurent series of $$f(z)= \dfrac{2sin(z)}{z^2-4} + \dfrac{cos(z)}{z-3i}$$ about $z=-2$ converges for $0<|z+2|<R$? Attempt : Its ...
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1answer
50 views

Find the taylor expansion to $(x^2 + x)e^{2x}$

My task is this: Find the taylor expansion to$$f(x)=(x^2 + x)e^{2x}.$$ My work so far: We should get $$e^{x}=\sum_{n=0}^\infty\frac{x^n}{n!}\implies e^{2x}=\sum_{n=0}^\infty\frac{(2x)^n}{n!}\...
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1answer
71 views

Finding taylor expansion of $\cos^2x$ and $\sin^2x$

My task is this: Find the taylor-series of $\cos^2x$ and $\sin^2x$. My work so far: We know that $\cos^2x \backslash \sin^2x = \frac{1\pm \cos 2x}{2}$, and the series for $\cos x = \sum_{n=0}^\...
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19 views

What is the name of this approximation?

I remember studying a while back about an approximation method where the error is calculated using $$ E_{n}=M_{n+1}-a_{n+1} \widetilde{T}_{n+1} $$ Where $\widetilde{T}_{n}=\frac{{T}_{n}}{2^{n-1}}$, ...
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1answer
28 views

A basic question about the decay rate of $te^{-t}$ as $t$ tends to infinity

It is well-known that $te^{-t}$ tends to $0$ as $t$ tends to infinity. But I want to know the decay rate of $te^{-t}$ as $t$ tends to infinity. Using Taylor expansion of $e^{t}$ we have: $${t /e^{t}}=...
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26 views

Range of values of $x$ for which the expansion $\ln(2+x)$ to valid

It is known that $$\ln(1+x) = \sum_{n=1}^\infty{(-1)^{n+1} \dfrac{x^n}{n}}$$ for $-1<x\leq1$. Question: What is the range of values of $x$ for which the expansion of $\ln(2+x)$ is valid? I ...
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2answers
53 views

How do you show that a power series of $e^x$ at a point converges to $e^x$?

My task is this: Find Taylor-expansion of $e^x$ at the point $1$, the convergence interval $I$ and then show that the series converges to $e^x$. My work so far: By using $Te^x @ x= 1$ we should get ...
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1answer
35 views

Taylor expansion of function [closed]

I try to figure out how the taylor expansion of the following function looks like, but so far I wasn't successfull: $z↦e^{iuz}−1−iuz$ for $|z|<1$. Who has an idea?
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18 views

Taylor series approximation of inverse trigonometric function

Suppose we have a function of three variables $a,b,c$ defined as, $f(x,y,z)=\arctan\left(\frac{\sqrt{x^2y^2-z^2}}{y^2-z}\right)$. Suppose $x=a, y=b, z=c$ satisfy the following property: (1) $a,b,c>...
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2answers
48 views

Find the fourth Taylor polynomial of f(x)=ln(x+1) at x=1

Let $f(x)=\ln(x+1)$ then (a) find the fourth Taylor polynomial of f at x=1 and (b) use part (a) find the approximate the value of ln(2.2) correct 4 decimal (c) Find an estimate for the error in ...
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Let $s(x)=\sum_{n=0}^\infty \frac1{(2n+1)!}x^{2n+1}$, $c(x)=\sum_{n=0}^\infty \frac1{(2n)!}x^{2n}$ prove that: $c(x)^2-s(x)^2=1$

Let $$s(x)=\sum_{n=0}^\infty \frac1{(2n+1)!}x^{2n+1}$$ $$c(x)=\sum_{n=0}^\infty \frac1{(2n)!}x^{2n}$$ prove that $$c(x)^2-s(x)^2=1$$ I know that the following series are representations of the cosh ...
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1answer
34 views

How to expand $x^n$ as $n \to 0$?

I am trying to expand $x^n$ in small $n$ using Taylor series. Using wolfram alpha, I found that it is $1+ n\log(x) + \cdots$ I tried to Taylor expand $x^n$ around $n=0$ but I cannot get this result.
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Interesting behavior of the expansion of $_1F_2(\alpha/2;3/2,\alpha/2+1;y^2/4)$ near $y=\infty$

When we use Mathematica 10.0 to expand generalized hypergeometric function $_1F_2(\alpha/2;3/2,1+\alpha/2;y^2/4)$ near $y=\infty$ with $\alpha$ a complex number, we obtain: $${_1F_2}(\alpha/2;3/2,1+\...
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908 views

Why the existence of Taylor series doesn't imply it coverges to the original function

Please note that I've read this question and it did not address mine. I've been presented with the following argument regarding Taylor series: We have a function $f(x)$, now assume that there ...
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1answer
24 views

Analytic and smooth functions

In my work, I first make an assumption: Assume the function $f(x)$ is an analytic function of $x$. Based on this assumption, I expand $f$ as Taylor series $$ f(x)=f_0+f_1x+f_2x^2+f_3x^3+\dots $$ ...
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26 views

Solve the following ODE using a Maclaurin expansion of the non-linear terms

Find two proper series solutions about the ordinary point $x=0$ of $$y''+e^xy'-y=0.$$ My proposed solution: Note that $e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!}.$ Assume there exists a power series ...
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1answer
21 views

Taylor and Macluarin series deriving

Hi to everyone Here i am studying Taylor series. $$f(x)=c_0 + c_1 (x-a) + c_2 (x-a)^2+ ...$$ $$ f(x)= f(a) + \frac{df(a)/dx}{1!}(x-a)^1 + \frac{d^2f(a)/dx^2}{2!}(x-a)^2 ...$$ Well my problem is ...
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16 views

Estimate an Taylor approximation II

i am doing some exercise for my numerical analysis course. And i found myself wondering if the following argument is legal. The context of this exercise is the smoothend newton algorithm, especially ...
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20 views

Laurent series about singular point for: $\left( \frac{1+x}{1-ax}\right)^{\frac{1}{3}}$

$\left( \frac{1+x}{1-ax}\right)^{\frac{1}{3}}$ I wish to find the Laurent series about the singular point $x=1/a$. I can find an expansion for the left side ($x=0$) and the right side ($x \rightarrow ...
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1answer
24 views

Taylor Series of a composition of functions

I have to find a Maclaurin series of the following function: $y = D\sin(C\arctan(Bx - E(Bx - \arctan(Bx)))) + Sh$ I wasn't able to find it by hand. Thanks in advance!
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1answer
45 views

Series expansion of infinite series raised to the $n$th power

So I know there is a well-known straightforward way to expand something like $$(a+b)^n$$ and that there are formulas which allow us to expand trinomials and multinomials in general. My question is, ...
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The remainder of taylor approximation, lagrange form of the remainder. The idea

We know that the formula for the remainder of taylor approximation is: $$R_n(x) = \frac{f(z)^{n+1} *(x-a)^{n+1}}{(n+1)!}$$ But also we have the formula: $$R_n(x) = \frac{M *(x-a)^{n+1}}{(n+1)!}$$ ...
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2answers
47 views

Find the Maclaurin series for $\cos^2(x)$

I am given this as a hint: $\cos^2(x) = \frac{1 + \cos(2x)}{2} \\$ I am not really sure how to start this one, would it just be the regular Maclaurin series squared. For example: $ (\sum_{n=0}^\...
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1answer
17 views

Relation between coefficients of two different power series.

Let $$f(z) = \sum_{n\geq 0} = a_nz^n, a_n\in\Bbb{C}$$ has a radius of convergence $\rho$. Then we can write $f(z) = \sum_{n\geq 0} b_n (z-\frac{\rho}{2})^n$ for $\{z: |z-\dfrac{\rho}{2}|<\dfrac{\...
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1answer
106 views

Approximate roots of nonlinear equation (non-integer polynomial)

In case of pulsating bubble arising from underwater explosion, bubble radius satisfies the following equation. $x^3\dot{x}^{2} + x^3 + \frac{k}{x^{3(\gamma-1)}} = 1$ The minimum and maximum bubble ...
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25 views

Mclaurin series and n-th derivative

(1) Find the general formula of the McLaurin series of $ f(x) = arctan((x^3)/2)/x^3\ $ (2) Evaluate the 18-th derivative of f(x) (3) Evaluate lim to infinity of f(x) By general formula do we just ...
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1answer
42 views

Taylor series doesn't seem to have a pattern?

My teacher gave us a study guide to work on, and one of the problems doesn't seem to come out right. The directions are to "find the Taylor series of $f(x)=x^5-3x^4+x^3+2x-1$ for $a=1$. I calculated ...
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1answer
33 views

Taylor series third order approximation

There has been this question that had been bothering me for a while and I could not find a satisfying answer on the internet or any of the books even though it is not very complex. i) Its because if ...
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1answer
41 views

Does multiplying Taylor series by an integer change the interval of validity.

If I have a Taylor series for example, $\frac{1}{1-x} = 1 + x + x^2 + x^3 + x^4 + \ldots, \qquad \text{valid for $-1<x<1$} $ and I multiply the series by some integer, let's say $5$, in ...
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1answer
35 views

If the derivative is written as shifts, can you relate it to the laplace/fourier tranform?

I was wondering if there is a way to write the derivative as an exponential? This might sound crazy at first, but I recently came across this formula for the Taylor expansion in three dimensions: $$\...
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1answer
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Taylor expansion of $f(x(t),y(t))$ around the point $(x_0,y_0)$.

My main question is basically whether the fact that both inputs depend on $t$ is an issue? Because if $x$ changes then $t$ must have changed and thus $y$ is likely to have changed. So would we need ...
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2answers
57 views

How do I show that $\ln(2) = \sum \limits_{n=1}^\infty \frac{1}{n2^n}$?

My task is this: Show that $$\ln(2) = \sum \limits_{n=1}^\infty \frac{1}{n2^n}$$ My work so far: If we approximate $\ln(x)$ around $x = 1$, we get: $\ln(x) = (x-1) - \frac{(x-1)^2}{2} + \frac{(x-...
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22 views

Solving limit of Integral through Taylor

Let $u:U\rightarrow \mathbb{R}$ ($U\subseteq \mathbb{R}^3$) be twice continously differentiable. Evaluate the limit: $$\lim_{r\to 0^+} \frac1{r^2} \Bigg( \frac1{4\pi} \iint\limits_{\xi^2+\eta^2+\...
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38 views

Can we Relate Radius of Convergence of Taylor Series and Asymptotic Rate of Growth?

I still need to be disabused of the belief that there is some simple connection between the finiteness of the radius of convergence and the asymptotic rate of growth. 1. Can we develop any ...
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Taylor series question help!

This question is on a past paper for my exam but no model solutions have been provided and I'm worried I'm doing completely the wrong thing, Consider two functions represented by Taylor (MacLaurin) ...
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1answer
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Could someone please confirm my answer this Maclaurin series??

Find three nonzero terms of the Maclaurin series of the function $f(x)={3/5} tan5x/x$ Using the maclaurin series i found them to be.. $3/5+x^2/25+2x^4/25$ Is this correct? If not what is the ...
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14 views

Determine the Lagrange Residual of $\ln(\frac{1-x}{1+x})$

Show, for $x_0=0$, that $\ln(\frac{1-x}{1+x})=-2\big[x+\frac{x^3}{3}+\dots+\frac{x^{2n-1}}{2n-1}+R_{2n}(f,0)(x)\big]$, with $$R_{2n}(f,0)(x)=-\frac{x^{2n+1}}{2n+1}\bigg(\frac{1}{(1+\theta x)^{2n+1}}+\...
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385 views

Fractional Derivative of a Taylor Series?

I have a function defined only by it's taylor series: $$f(x) = \sum_{k=0}^\infty \frac{f^{(k)}(0)}{k!}x^k$$ Obviously, integer derivatives can be defined as $$\frac{d^n}{dx^n} f(x) = \sum_{k=0}^\...
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42 views

Consider the function $f(x) = e^{x^2}\ln(1+x)$ for $0 < x < 1$

So I was able to do the first half of this problem (part a), which was: $$e^{x^2}\ln(1+x) \approx x - \frac{x^2}{2} + \frac{4x^3}{3}$$ but I'm confused what my next step should be, for solving (part ...
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1answer
39 views

Factorization of Taylor series.

I know that for a (finite) polynomial $P(x) = a_nx^n + a_{n - 1}x^{n - 1} + \cdots + a_0$ whose zeros are $x_1, x_2, \ldots, x_n$, then we can factorize it as $$P(x) = a_n(x - x_1)(x - x_2) \cdots (x -...
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2answers
47 views

Find a function f so that Taylor expansion is always accurate to this degree

Find a function $f$ from R to N such that with $T$ be the Taylor expansion of $\sin(x)$ around $0$. $ | \sin (x) - T_{f(x)}x$| $\leq 1$ The hint is to use $n! \leq 3 \sqrt{n} {(\frac{n}{e})}^n$
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1answer
21 views

Is Every (Real) Analytic Function (with Non-Degenerate MacLaurin Series) Asymptotically Greater Than any Polynomial?

Question: Given a function $f: \mathbb{R} \to \mathbb{R}$ such that the MacLaurin series exists and equals the function for every $x \in \mathbb{R}$, and such that for all $n \ge n_0$, $n_0$ some ...
3
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0answers
39 views

Taylor series Lagrange Remainder explanation

So, given a Taylor series: $$f(x)=f(x_0)+f'(x_0)(x-x_0)+f''(x_0)\frac{(x-x_0)^2}{2!}+\cdot\cdot\cdot+f^{(n)}(x_0)\frac{(x-x_0)^n}{n!}+R_n$$ The error $R_n$ is given by: $$R_n=\frac{f^{(n+1)}(\xi)}{(n+...