2
votes
1answer
42 views

Showing that if the $n$th derivative of a function is bounded then it is real analytic

I reproduce from my lecture notes: Suppose $f$ is $C^\infty$ on $[a,b]$ with $$\left|f^{(n)}(x)\right|\leqslant M~~\text{for all}~~x\in(a,b).$$ Then $f$ is real analytic in $[a,b]$. Proof. ...
0
votes
0answers
27 views

Product of two $2$-variables Taylor series

Using the standard multi-index notation, suppose we have the two Taylor series $$ f(\theta) := \sum_{|\alpha|=0}^{\infty} a_{\alpha} \theta^{\alpha} $$ and $$ g(\theta) := \sum_{|\alpha|=0}^{\infty} ...
1
vote
1answer
37 views

What can be said about an infinitely differentiable function whose Taylor series diverges?

What can be said in general of an infinitely smooth function whose Taylor series diverges? According to Borel Theorem it is possible to construct such but what kind of property have those special ...
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
15 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
78 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
72 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
19 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
66 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
26 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
26 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
120 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
38 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
66 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
39 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
35 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
24 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
35 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
48 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
58 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
97 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
75 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
203 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
73 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
35 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
41 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
115 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
56 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
101 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
160 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
139 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
263 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
287 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
104 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
475 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
765 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
67 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
120 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 ...