1
vote
0answers
61 views

To show a power series is a Taylor series

Is it possible to prove if $f(x) = \sum_{n = 0}^\infty a_n(x - a)^n$ then the series is the Taylor series of $f$ without using complex analysis, as done here?
0
votes
0answers
10 views

Taylors formula

I have a circle $K(a,\epsilon) \subset \Omega $ and for $ \parallel \Delta x \parallel \lt \epsilon $ we look at $ \Delta f = f(a+ \Delta x) - f(a) $ Now I look at the function $ F:[0,1]\rightarrow ...
1
vote
4answers
77 views

If $\displaystyle \sum_{n=0}^{\infty}c_{n}x^{n}=0$, show that $c_n= 0$ for any $n$ [closed]

Suppose that $f(x)=\displaystyle\sum_{n=0}^{\infty}c_{n}x^{n}$ for all $x$ with the radius of convergence $R>0$. If $\displaystyle\sum_{n=0}^{\infty}c_{n}x^{n}=0$, show that $c_n=0$ for any $n$.
1
vote
2answers
62 views

3rd Order Taylor expansion of $e^x\cos(y)\sin(z)$

I'm looking for the 3rd.-order Taylor approximation of $(x,y,z) \mapsto e^x\cos(y)\sin(z)$ at $(x_0,y_0,z_0) = (0,0,0)$ I've got this piece of advice at hand: $\quad\textit{Use the Taylor series ...
1
vote
1answer
18 views

Linear functional vs. map

A few days ago we were briefly discussing Taylor's theorem in higher dimensions in the lecture. Referring to the expression $f(x)=f(a)+Df(a)(x-a)+$higher order the lecturer said that in general $Df$ ...
1
vote
1answer
47 views

Where to stop a taylor expansion of a function of more than one variable?

I tried to taylor expand a function of three variables $f(x,y,z)$ around $y=0,z=0$ but I don't find a way to decide where to stop the expansion. Usually when one expands a function of one variable ...
0
votes
0answers
64 views

Taylor theorem remainder term

I'm having trouble applying the formula for the remainder in the Taylor's theorem. From Wikipedia we know that for $f(x)=f(a)+f'(a)(x-a)+…\frac{f^{(n)}(a)}{n!}(x-a)^{n}+R$ the remainder $R$ in the ...
0
votes
2answers
28 views

Complex functions and Taylor series

Find the Taylor series arround $z_0=0$ write radius of convergence a) $f(z)=\cosh(z)$ b) $f(z)=\log(z+1)$ I don't know how it works with the complex functions. Could you show me the workflow? I ...
0
votes
0answers
25 views

Finding a function with a given Taylor expansion

Is there any function $f(x)$ which has the following Taylor series representation? $$ f(x) = \sum_{k=0}^{\infty}{c_{k} (1 + \frac{2x^{2}}{k})^{-k/2}}. $$ for some coefficients ...
1
vote
2answers
25 views

Taylor theorem and a $C^{3}$ function with the following property…

A function $f:\mathbb{R} \rightarrow \mathbb{R}$ is $C^3$ with $$f(a+h)=f(a)+f'\left(a+\dfrac{1}{2}h\right)h$$ whenever $a \in \mathbb{R}$ and $h \geq 0$. By applying Taylors Theorem to $f$ and to ...
5
votes
2answers
118 views

Remainder form of Taylor polynomial at $x_0$: $ \frac{1}{n!}f^{(n)}(x_0 +\theta(x-x_0))(x-x_0)^n$ with $\theta \to \frac{1}{n+1}$ as $ x \to x_0$

If the function $f: \mathbb R \to \mathbb R$ is $n+1$ times differentiable at $x_0$ and $f^{(n+1)}(x_0) \neq 0$, then a form of the remainder in Taylor's Formula is supposedly $$r_n(x_0;x) = ...
0
votes
2answers
37 views

Using Taylor's Theorem to show $|x-tan(x)|\leq 1/300$ for $0\leq x \leq 1/10$

Using Taylor's Theorem deduce that for $0\leq x \leq 1/10$ $|x-tan(x)|\leq 1/300$ So my attempt; to get the taylors theorem about $x_0=0$ $f(x)=x-tan(x)$ $f'(x)=1-sec^{2}(x)$ ...
1
vote
1answer
65 views

Is there closed form for $(1-p)(1-p^2)(1-p^3)…$ or its Taylor expansion?

I was considering the following problem: Somebody uses a backup for something (e.g. backups a file) and the backup is equally reliable as original storage. The storage is not perfectly reliable and ...
0
votes
0answers
15 views

Differentiable function made up of arbitrary points.

Hi all, for this question , my attempt so far is; The function $F$ here is considered as a function of $t$ alone; the value of $x$ is regarded as a constant. Of course, if we change the value of $x$ ...
1
vote
1answer
36 views

Proof concerning logs and taylor series

Prove that if $n$ is a positive integer and $|x| \leq \dfrac{1}{2}n$ then $(i)\quad n\log\left(1+\dfrac{x}{n}\right)=x+Q_{n}(x)$ where $(ii)\quad |Q_{n}(x)|\leq\dfrac{|x|^{2}}{n}$ and deduce ...
2
votes
1answer
34 views

A question on Taylor expansion/approximation

Suppose we are given a continuos function $f(x)$ where $x \in [0,2]$, and the function $f(x)$ is $n$-th-order differentiable, for $n \in \mathbb{N}$ and $n>2$. Besides, we know that these ...
1
vote
0answers
23 views

Proof of lagrange inversion of taylor series

is there a proof for the lagrange inversion of taylor series? The formula is given in http://en.wikipedia.org/wiki/Lagrange_inversion_theorem#Theorem_statement The proof cannot be found in the ...
6
votes
0answers
52 views

Higher Order Derivative Proof .

I would appreciate if someone could check over my proof for this question and advise me if it is correct. My attempt so far; Now as $f$ is k times differentiable , it taylor series about $x_{0}$ ...
1
vote
1answer
19 views

Compostion of functions in taylor series

I was attempting to help answer a question, but I am curious about the following. Suppose we have an analytic function $f(z)\equiv\sum\limits_{k=0}^{\infty} a_k(z-z_0)^k $ on some suitable disk of ...
1
vote
1answer
33 views

differential equation as taylor series

Consider the equation $\frac{\mathrm{d} x(t)}{\mathrm{d}t} = g(x(t))$ , with $x(0) = x_0$, where g is function that admits derivatives of all orders.If the solution of the equation can be written as a ...
1
vote
1answer
40 views

Is there a more concise way?

I am currently trying to prove Taylor's Theorem, this is the proof I am given: I am finding this very long and hard to follow, can anyone give me a more concise, clear proof? Thanks
0
votes
3answers
53 views

Using Taylor expansion to evaluate infinite sum

How do I use the Taylor expansion of $$(1+x)^{-\frac{1}{2}} $$ to evaluate $$ \sum_{n=0}^{\infty}\binom{2n}{n}\left(-\dfrac{6}{25}\right)^{n} $$ Thanks
1
vote
3answers
46 views

Expand a sin(x^3) in Maclaurin's series and find a 30th derivative at (0)

I have a big task and problems with it. I have to expand this function in Maclaurin's series. $$cos(x^{3})$$ I tried expand it as $$\sum_{n=0}^\infty (x^n)/n!$$ but it's for $cos(x)$. So i don't know ...
1
vote
0answers
19 views

Expand a function in Maclaurin's series

Please help me expand this function in Maclaurin's series to find an interval of convergence. I tried to turn it into this expression: but it doesn't give me the result. Help me please it's ...
1
vote
1answer
40 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 ...
1
vote
1answer
56 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 ...
1
vote
0answers
95 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 ...
0
votes
0answers
59 views

How many terms to use in a Taylor series for local truncation error

So from my understanding for a finite difference approximation, you're supposed to expand the series "about" the point $x$, e.g., $$u(x+h) = u(x) + h \ u'(x) + ...
4
votes
2answers
196 views

Taylor series of the inverse of $x^4+x$

I would like to expand the inverse function of $$g(x) := x^4+x $$ in a taylor series at the point x = 0. I calculated the first and second derivate at x = 0 with the rule of the derivation of an ...
3
votes
2answers
71 views

Remainder of Taylor series

The Taylor series of the function $$f(x) = \int_{1}^{\sqrt{x}} \ln(xy)+ y^{3x} dy + e^{2x}$$ at the point $x = 1$ is $$e^2 + (x-1)\left(2e^2+\frac{1}{2}\right) + ...
0
votes
1answer
34 views

Tangent and Taylor polynomials

We know that this series $x+ \frac{x^3}{3}+\frac{2x^5}{15}+\ldots$ is convergent in $|x|\lt \pi/2$, furthermore it converges to $\tan(x)$. I would like to know if we restrict to finite terms of this ...
1
vote
1answer
39 views

f is a smooth function, and $M_n$ is the sup of $f^{(n)}$. Show if $\lim_{n \to \infty} \frac{M_n}{n!}R^n < \infty$, then f(x) is the taylor series.

Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be a smooth function (i.e. assume that the n-th derivative $f^{(n)}$ is defined on all of $\mathbb{R}$). Let $R$ denote the radius of convergence of the Taylor ...
4
votes
3answers
114 views

How to check the real analyticity of a function?

I recently learnt Taylor series in my class. I would like to know how is to possible to distinguish whether a function is real-analytic or not. First thing to check is if it is smooth. But how can I ...
1
vote
1answer
55 views

Weierstraß approximation on the real line

First of all: I am aware of the thread Weierstrass approximation does not hold on the entire Real Line. My question is just that if we have a function like $sin(x)$ that can be approximated by its ...
2
votes
1answer
101 views

Evaluate $I=\int_{\gamma (0,1)}\frac{zdz}{(z^2-4z+1)^2}$ using Taylor's Theorem

I've shown that $f(z)=\frac{z}{(z^2-4z+1)^2}$ is holomorphic apart from at points $\alpha=2-\sqrt3$ and $\beta=2+\sqrt3$ and that the talyor coefficient of $g(z)=\frac{z}{(z-\beta)^2}$ centred at ...
1
vote
0answers
98 views

The remainder of Taylor (Maclaurin) series of $\cos(x)$

Something is bothering me with the remainder of the Taylor (Maclaurin) series of $\cos(x)$. The formula of $a_n$ is $(-1)^k \frac{x^{2n}}{(2n)!}$. By Leibniz Theorem, $r_n<a_{n+1}$ which is, ...
2
votes
1answer
157 views

Complex Analysis - Taylor Series

The question reads as follows: Let the function $f$ be given by $f(z)=\frac{z}{2}+\frac{z}{e^z-1}$ if $z\neq0$ and $f(z)=1$ if $z=0$. Show that $f$ is analytic at $z=0$ and that $f(z)=f(-z)$. ...
3
votes
4answers
137 views

Taylor series of $\arctan(x+2)$ at $x=\infty$

The simple question is: what is the correct way to calculate the series expansion of $\arctan(x+2)$ at $x=\infty$ without strange (and maybe wrong) tricks? Read further only if you want more details. ...
1
vote
2answers
248 views

Short way? Write taylor's formula $f(x,y)=e^{xy}$ $a=(0,0)$ and $p=4$

Write taylor's formula $f(x,y)=e^{xy}$ $a=(0,0)$ and $p=4$ Does there exist any short way? I have to calculate all partial dervatives. Is it?
2
votes
1answer
284 views

Taylor remainder of $f(x,y)=\sin x\cdot \cos y$

Given $f\colon \mathbb R^2\rightarrow\mathbb R,(x,y)\mapsto\sin x\cdot\cos y$ I want to show that there exists $M>0$ such that $$|f(x,y)-T_2(x,y)|\leq M(|x|+|y|)$$ for all $(x,y)\in\mathbb R^2$. ...
3
votes
1answer
103 views

Differentiate a hypergeometric function expression

I have the following function $$f_\epsilon (p)=\frac{1}{2}(1-p)^\epsilon 2^\epsilon {_2}F_1(1-\epsilon,\epsilon;1+\epsilon;\frac{1-p}{2}),\qquad p\in(-1,1).$$ Here $F$ is the hypergeometric ...
4
votes
4answers
460 views

Find nth derivative of $\frac{x^{n}}{(1-x)^{2}}$, please?

I need to find the nth derivative of $\frac{x^{n}}{(1-x)^{2}}$ for $0<x<1$ So far, I tried the same method used for $\frac{x^{n}}{1-x}$ and here's what I got: \begin{equation} ...
0
votes
1answer
29 views

Prove that the polynomial divided by a fraction of the power of n is equal to the sum of fractions of any constans and successive powers of

Let n≥1 and n is integer. P(x) - polynomial and $deg P(x)<n$. Prove if $ a \in \Bbb R/{0} $ then: $ \frac{P(x)}{(ax+b)^n} = \frac{c_1}{ax+b} + \frac{c_2}{(ax+b)^2}+...+\frac{c_n}{(ax+b)^n}$ for ...
0
votes
1answer
732 views

estimating the error of $\sin(x) = x$ with Taylor's Theorem

I want to calculate the numerical error in approximating $\sin(x)=x$ with Taylor's Theorem. Furthermore, what values of $x$ will this approximation be correct to within $7$ decimal places? Here is ...
1
vote
1answer
66 views

Estimate the scale of the power series with Poisson pdf-like terms

Sorry to bother you, but I guess that this question is not appropriate for MO, so I repost it here hoping that someone could give me a clue. I would like to have an estimate for the series $$P(t) = ...
1
vote
0answers
119 views

Taylor Expansion of Power Series

Suppose that $\space f:[0,1]\rightarrow \mathbb{R}$ is real analytic and that its power series expansion is: $\\ f(x)=\sum\limits_{n=0}^\infty a_nx^n$ Prove that there exists an $x_0\epsilon (0,x)$ ...
1
vote
1answer
66 views

Inverse Function Thorem

Let $f,g:\mathbb R\to\mathbb R$ be smooth functions with $f(0)=0$ and f'$(0)\neq 0$. Consider the equation $f(x)=tg(x), t\in \mathbb R$. Show that in a suitable small interval $|t|\leq \delta$, there ...
5
votes
1answer
157 views

Taylor series and tempered distributions

Suppose we have a function $\psi$ in $\mathbb{R}$ other than a polynomial which is equal to its Taylor Series for every point in $\mathbb{R}$. Why is the following statement valid? When we interpret ...
1
vote
1answer
146 views

Derivative of a little-o remainder

If we have a $\phi: \mathbb{R} \times \mathbb{R}^n \times \mathbb{R} \to \mathbb{R}$, $\phi = \phi(t, \mathbf{q},\alpha)$ one-parameter group of infinitesimal transformation which is $\mathcal{C}^2$ ...
1
vote
1answer
451 views

Curvature via hessian in Taylor expansion

In the case of a univariate function, the smaller the second derivative in its Taylor expansion, the smaller is the curvature of the univariate function. Now, how is the curvature of the function ...