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.

learn more… | top users | synonyms (2)

0
votes
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!}\...
1
vote
1answer
72 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}^\...
2
votes
0answers
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 $\widetilde{T}_{n}=\frac{{T}_{n}}{2^{n-1}}$, ...
0
votes
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}}=...
0
votes
0answers
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 ...
2
votes
2answers
54 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 ...
0
votes
1answer
59 views

Explain inequality of integrals by taylor expansion

I try to understand why the following inequality holds. $$\left|\int_{|y|<1} e^{iuy}−1−iuy\ \, dy \right| \le \frac{1}{2} \cdot \int_{|y|<1} |uy|^2\ \, dy$$ Due to a hint I'm pretty sure, ...
-1
votes
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?
2
votes
1answer
29 views

Local quadratic approximation

I wanted to implement some penalized regression parameter estimation algorithm by Fan&Li (http://sites.stat.psu.edu/~rli/research/penlike.pdf, section 3.3, [1]), but cannot catch the idea of some ...
1
vote
2answers
66 views

Induction Proof of Taylor Series Formula

I'm attempting to prove a formula for the taylor series of function from a differential equation. The equation is $$f(0)=1$$ $$f'(x) = 2xf(x)$$ I have found empirically that $$f(x) = \sum_{k=0}^{\...
0
votes
0answers
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>...
0
votes
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 ...
1
vote
2answers
35 views

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 ...
1
vote
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.
2
votes
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}\...
9
votes
6answers
909 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 ...
0
votes
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 $$ f(x)=f_0+f_1x+f_2x^2+f_3x^3+\dots $$ ...
10
votes
2answers
324 views

Closed form for $ S(m) = \sum_{n=1}^\infty \frac{2^n \cdot n^m}{\binom{2n}n} $ for integer $m$?

What is the (simple) closed form for $\large \displaystyle S(m) = \sum_{n=1}^\infty \dfrac{2^n \cdot n^m}{\binom{2n}n} $ for integer $m$? Notation: $ \dbinom{2n}n $ denotes the central binomial ...
0
votes
0answers
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 ...
2
votes
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 ...
0
votes
0answers
17 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 ...
0
votes
1answer
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 ...
1
vote
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, ...
0
votes
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!
0
votes
0answers
11 views

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: $$R_n(x) = \frac{M *(x-a)^{n+1}}{(n+1)!}$$ ...
1
vote
1answer
45 views

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+\...
0
votes
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}^\...
0
votes
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{\...
-1
votes
1answer
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 ...
0
votes
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 ...
0
votes
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 ...
0
votes
1answer
18 views

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 ...
1
vote
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-...
0
votes
0answers
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+\...
0
votes
0answers
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}}+\...
0
votes
0answers
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 ...
1
vote
1answer
40 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 -...
0
votes
2answers
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 ...
1
vote
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
votes
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+...
0
votes
1answer
19 views

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 ...
0
votes
2answers
27 views

Could some confirm my answer for this limit using taylor series?

$\lim_{x→0}$ $\dfrac{x^2}{x\sqrt{1+x} −\ln(1+x)}= ?$ I got $-2$. Is this correct if not what is the answer so i can find out where i went wrong. Thanks in advance
3
votes
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 $n! \leq 3 \sqrt{n} {(\frac{n}{e})}^n$
1
vote
0answers
57 views

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) ...
0
votes
0answers
22 views

Taylor expansion in proof of weak maximum principle

Picture below is part of proof of weak maximum principle. On the red line ,I don't know how to use the Taylor expansion to get $-u''(x_0) \le 0$. I think the Taylor expansion of $u(x)$ at $x_0$ is $$ ...
1
vote
0answers
19 views

Taylor series roots at infinity

I started thinking about this after this MathSE thread. Take a sequence of Taylor polynomials $f_n$ that converge to $f$. Does $f_n$ always have a growing number or roots in $\mathbb{C}$ which grow ...
1
vote
3answers
42 views

Complex analysis: Using Taylor expansion to show $|c_n| ≤ \frac{1}{r^n}\sup_{z∈C_r(0)}|f(z)|$

Consider the function $f$ is defined through the power series $$f(z) := c_0 + \sum_{n=1}^\infty c_nz^n$$ and assume that the series on the right has a radius of convergence $R > 0$. Show that $$|...
0
votes
1answer
32 views

Limit calculate using Maclaurin series

I need help to calculate this limit using Maclaurin series: $\lim_{x\to \infty}((x^3-x^2+\frac{2}{x})e^{\frac{1}{x}}-\sqrt{x^3+x^6})$ I don't know from where to start. I think I need to to write ...
0
votes
1answer
23 views

Polynomial approximation of a limit

I am supposed to find the Taylor polynomial $P_2(x;1)$ for the exponent function $f(x)=e^x$ and use it in conjunction with Taylor's theorem to evaluate the following limit: $$\lim_{x\rightarrow1} \...
1
vote
1answer
31 views

$\lim_{x \to 0}\frac{(x^2 \times 2^x \times (\log 2)^2) - (2^x - 1)^2}{(2^x - 1)^2(x^2 \times \log 2)} = ?$

$$\lim_{x \to 0}\frac{(x^2 \times 2^x \times (\log 2)^2) - (2^x - 1)^2}{(2^x - 1)^2(x^2 \times \log 2)}$$ I tried this by using the Taylor series $2^x = 1 + x\log 2 + \frac{x^2}{2!}(\log2)^2 + \dots$....