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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23 views

Using Taylor's series in imporper integrals

Is it possible to simplify an improper integral using Taylor's series? How can I prove this procedure is correct? For example, take $$f(\alpha)=\int_0^{\infty} ...
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3answers
114 views

A system of $n$ equations , how does it behave for growing $n$?

I read about the system of $n$ equations in the link below. I wonder how it behaves for growing $n$. Does it converge ? http://math.eretrandre.org/tetrationforum/showthread.php?tid=889 Here it is ...
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1answer
51 views

Maclaurin series of (1+x)^(1/x)

how can i find the Maclaurin series of $f(x)=(1+x)^{1 \over x}$? $f(0)$ is not even defined, or should I define it as $f(0)=e$? I stopped at the first derivative as it gets terribly messy. thank ...
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1answer
224 views

Real-analytic periodic $f(z)$ that has more than 50 % of the derivatives positive?

Im looking for a real-analytic function $f(z)$ such that for any $z$ $1) $$f(z+p) =f(z)$ With $p$ a nonzero real number and where $z$ is close to , or onto the real line such that $z$ is in the ...
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2answers
62 views

The simplest way to pow using only simple arithmetic

i want to get function $f(x, a) = x^a$, for both x and a - real numbers, that uses only + - * /. So only way I found is: get taylor series for $$x^a = \sum_{n = ...
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0answers
18 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 ...
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1answer
37 views

Taylor series $\ln(2+x)$ centered at $x=2$

Taylor series $\ln(2+x)$ centered at $x=2$. Is the correct result $$y=\ln \left(4\right)+\sum _{n=1}^{∞}\frac{\left(-1\right)^n}{4^{\left(2^{\Large n}\right)}}\cdot \frac{\left(x-2\right)^n}{n!}\ ?$$ ...
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2answers
33 views

Meaning of interval of convergence when approximating functions

Let's say I have a Taylor series approximation, $p(x)$, of a function $f(x)$ at $a$: $$ p(x)=\sum_{n=0}^\infty{\frac{f^{(n)}(a)}{n!}(x-a)^n} $$ And that this Taylor series has a radius of ...
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0answers
30 views

Taylor expansions, inequalities and more

(Part A) I have to find the Taylor expansion of order 2 around (0,0) of $$f: \mathbb{R}^{2}\rightarrow \mathbb{R}$$ $$(x,y)x \mapsto f(x,y) = x\log (1+y)+sin(x+y) $$ Furthermore I have to prove if ...
2
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1answer
60 views

Finding $\zeta(4)$ by Taylor series

Is it possible to solve Zeta(4) function using something similar to the solution for zeta(2) as seen in this video? https://www.youtube.com/watch?v=mTPKyC3Udns
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2answers
121 views

Proving $\forall x\in\mathbb R : \dfrac{e^x + e^{-x}}2 \le e^{\frac{x^2}{2}}$ with Cauchy's MVT

Prove for all $x\in\mathbb R$: $$\dfrac{e^x + e^{-x}}2 \le e^{\frac{x^2}{2}}$$ Mclauren expansion: $$e^x=1+x+\frac{x^2}{2}+\frac{x^3}{3!}+R_4(x)$$ ...
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0answers
20 views

solution check for approximating derivative using a Taylor expansion.

I'm wondering if there's in a mistake in either my reasoning or the given solution for the problem and was hoping to have someone double check this for me. The problem states: Let $g(2)=3$ , ...
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1answer
16 views

taylor series please i need immediate help

Expand 1/z by Taylor series about a point z=1. what I have done really makes no sense because I have no idea about it. I can only think of 1/z=1/z-1+1 1/z=1/((z-1)*(1/z-1))
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3answers
68 views

When does the remainder term in the taylor series go to zero?

When does the remainder term in the taylor series go to zero? Theorem: Let $f\in C^{N+1}([\alpha,\beta])$ and $x,x_0\in(\alpha,\beta)$. Then ...
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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 $n$.
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2answers
58 views

Taylor series of $\sin(x)$ converges uniformly on $[-\pi,\pi]$?

According to my notes, the Taylor series of $\sin(x)$ converges uniformly on $[-\pi,\pi]$. I know that the remainder term needs to converge uniformly to $0$ for this to be the case. But I really ...
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1answer
51 views

How can I find the Maclaurin series of $\frac{1}{1+x+x^2}$?

How can I find the Maclaurin series of $\frac{1}{1+x+x^2}$? At first, I found the Maclaurin series of $\frac{1}{1+x}$, which is $\sum_{n=0}^{\infty}(-1)^{n}x^{n}$ and simply replaced $x$ with $x^2 + x ...
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2answers
57 views

$f(x)=\begin{cases}e^{\frac{-1}{x^2}} & \text{ if } x\neq 0 \\ 0& \text{ if } x= 0\end{cases}$ is not equal to its Maclaurin Series

$f(x)=\begin{cases}e^{\frac{-1}{x^2}} & \text{ if } x\neq 0 \\ 0& \text{ if } x= 0\end{cases}$ is not equal to its Maclaurin Series, which is ...
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1answer
29 views

Expanding $\frac{1}{\sqrt{1-x^2}}$ through the expansion of $\frac{1}{\sqrt{1-x}}$ by binomial series

I heard that to expand $\frac{1}{\sqrt{1-x^2}}$, I have to expand $\frac{1}{\sqrt{1-x}}$ by binomial series and then just replace $x$ to $x^2$. Using binomial series, I found that ...
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3answers
31 views

How can I expand $\frac{1}{\sqrt{1-x^2}}$ by using the binomial series?

How can I expand $\frac{1}{\sqrt{1-x^2}}$ by using the binomial series? I know how to expand $\frac{1}{\sqrt{1-x}}$, but I have no idea how to expand $\frac{1}{\sqrt{1-x^2}}$. Simply differentiate ...
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1answer
40 views

Cubic MacLaurin $e^{x^2}$

Find the Cubic MacLurin expansion of e^{x^2}. First, I tried the sub $t=x^2$ and used the regular expansion for $e^t$. But that was wrong. Can I not do non-linear substituions? My calculations: ...
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4answers
136 views

Two ways to show that $\sin x -x +\frac {x^3}{3!}-\frac {x^5}{5!}< 0$

Show that: $\large \sin x -x +\frac {x^3}{3!}-\frac {x^5}{5!}< 0$ on: $0<x<\frac {\pi}2$ I tried to solve it in two ways and got a little stuck: One way is to use Cauchy's MVT, define $f,g$ ...
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3answers
30 views

Maclaurin series stuck at finding $L_n$

I need to develop Maclaurin serie of $f(x)=\frac{1}{(1-x)^2}$ I found all the derivative, and all the zero values for the derivatives. I come up with that : ...
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0answers
47 views

Understanding Taylor's Theorem

In our real analysis course, our lecturer has given us the following theorem, which I don't quite understand. It's been given in an odd way, not similar to anything I've found in books or on the ...
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2answers
40 views

How to show that $\cos(x)=\sum\limits_{n=0}^\infty (-1)^n \frac {x^{2n}}{ (2n)!} $

We know that cos(x) is infinitely differentiable and the Lagrange remainder $\rightarrow 0$ for all $x$, so the Taylor series indeed produces the function. We also know that $\cos^{(4k)}(x)=\cos(x)$ ...
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1answer
16 views

How can I find the Taylor series of a function using the known Taylor series of a related function?

I am trying to calculate the Taylor series for the function: $$f(x) = {\frac 1 x}(1 - \cos\sqrt{x})$$ How do I do it, if I know the Taylor series for $\cos(x)$? $\cos x = {\Large \sum\limits_{k = ...
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2answers
61 views

compute taylor series about $x=0$ of $\arctan(e^x -1 )$

hello I am having some issue and need a little guidance with this taylor expansion $$f(x)=arctan(e^x -1)$$ the terms i should get are $x+\frac{x^2}{2}-\frac{x^3}{6}-\frac{11 x^4}{24}-\frac{5 ...
2
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3answers
81 views

Taylor series of $\sqrt{\ln\left(\frac{1}{x}\right)}$

I am trying to compute the Taylor series of: $\sqrt{\ln\left(\frac{1}{x}\right)}$ I have computed the derivatives and evaluated them in $x=1/e$ but I cannot find the formula for the sequence of the ...
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3answers
75 views

Taylor expansion at $x=0$ of $\ln(1/(1-x))$

Hello I am having some trouble with the taylor expansion of $$f(x)= \ln \frac1{1-x}$$ Would it be correct to treat the inner part as the following geometric series? ...
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5answers
104 views

Taylor series for $\sinh1$

I am doing taylor series and I want to do it on $\sinh1$. is there a way to make this problem really simple before I begin? note: $\sinh x= \cfrac{e^x - e^{-x}}2$ Any ideas are really helpful ...
2
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2answers
53 views

Trying to solve a Taylor series problem

I have a Taylor series problem, well more precisely a Maclaurin series. I am trying to find convergence of: $f(x) = e^{x^3} + e^{{2x}^3}$ Okay here goes: $$f'(x) = 3xe^{x^3} + 6x e^{{2x}^3}$$ ...
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0answers
10 views

approximate a function by linear combination of its asympototics with fractional argument

Suppose that $0\le a \le 1$ and $0<f(x), g(ax)<\infty$ for $x\ge 0$ and $f(x)\to g(x)$ when $x\to \infty$. And also $g(a x)<g(b x)$ if $a<b$. Question: Can we use $g(a_k x)$ $(0\le a_1\lt ...
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1answer
30 views

Approximating a curve with a parabole at a given point

I wonder if such a task is possible: we have a curve defined implicitly with: $x^4+y^3-xy-1=0\qquad(1)$ I want to find a parabola in a general form of: $Ax^2+Bxy+Cy^2+Dx+Ey+F=0$ ... which ...
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2answers
86 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 ...
3
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1answer
26 views

Reference Request: Vector valued Taylor formula

It is well known that for analytic function defined on interval $I$ we have $$ f(x)=\sum^{\infty}_{k=0}f^{k}(0)\frac{x^{k}}{k!} $$ and for function defined on $I^{n}\rightarrow \mathbb{R}$ we have $$ ...
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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 $Df$ ...
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2answers
35 views

Evaluating a taylor series around a given point

So I'm having some trouble with the problem: Given that $\ln(x+1)=\sum_{n=1}^{\infty } \frac{(-1)^{n+1}}{n}x^{n}, -1<x\leq 1$, find the Taylor series of ln(x) around 3. For which x is this series ...
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1answer
77 views

Quick question on convergence

Shouldn't the radius of convergence be defined as: $$\frac{1}{R} = \lim_{n\rightarrow \infty} \left(a_n \right)^{\frac{1}{n}}$$ Not sure what they are doing above..
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60 views

Taylor series approximation using for a pdf

I have an question which links Taylor series to expectation and variance, but I'm really not sure what it's asking me to do. X is exponential with rate 1. The question asks me to use a three term ...
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2answers
73 views

Taylor Series Representation of $e^{1-\cos(x)}$

Hello I was wondering how to simplify this Taylor Series $$ e^{1-\cos(x)} =\sum_{k=0}^\infty\frac{(1-\cos(x))}{k!} ^k\ $$ to where I can write out the first couple of terms which are $ ...
3
votes
1answer
67 views

gradient of norm square of a random vector

Let $g(w)= \|Y_n - f(w,X_n) \|^2$ where $f:\Bbb R^d \times \Bbb R^m \to \Bbb R^k : w \in \Bbb R^d$. What is the gradient of $g$ ? $X_n$ and $Y_n$ are random vectors. Basically, I want to find ...
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1answer
19 views

Numerical Differentiation Given Set Of Values

Given the values $f(0),f(h),f(2h)$ and $f'(h)$ , I need to find a numerical differentiation of highest approximation order to approximate $f''(0)$. Usually I'd use Taylor expansion , but I need to ...
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1answer
64 views

$f(x)=e^x$ be approximated by Taylor's polynomial of degree $n$ at the point …

I am stuck on the following problem: Let $f(x)=e^x$ be approximated by Taylor's polynomial of degree $n$ at the point $x=\frac12$ and on the entire interval $[0,1]$. If the absolute error in this ...
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1answer
49 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 ...
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2answers
97 views

Show that $f(x)=e^x$

In this case $f(x)=1+x+x^2/2!+x^3/3!+x^4/4! + ... = \sum_{n=0}^\infty \frac{x^n}{n!}$. I understand it conceptually in terms of the Taylor series, but I have no idea how to prove it rigorously.
3
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3answers
204 views

Limit of $\dfrac{\tan^{-1}(\sin^{-1}(x))-\sin^{-1}(\tan^{-1}(x))}{\tan(\sin(x))-\sin(\tan(x))}$ as $x \rightarrow 0$

Find $\lim_{x \to 0} \dfrac{\tan^{-1}(\sin^{-1}(x))-\sin^{-1}(\tan^{-1}(x))}{\tan(\sin(x))-\sin(\tan(x))}$ I came across this limit a long time ago and could easily obtain a straightforward solution ...
2
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1answer
55 views

Show that $|f'(x)| \le \frac{2M_0}{h} +\frac{hM_2}{2}$ and $M_1 \le 2\sqrt{M_0M_2}$

Let $f$ be a twice derivable function and $M_i =\sup_{x \in \mathbb{R}} |f^{(i)}(x)|$ and $|M_0|, |M_2|<\infty $. Preferably using the Taylor series on the interval $[x,x+h]$ show the following ...
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0answers
14 views

Why more smooth the function the more precise finite difference method?

As the title, Why more smooth the function the better finite difference method? I guess that if the function is smooth we can better approximate with Taylor series, but formally how this helps? ...
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1answer
31 views

Finding an upper bound error of a Maclaurin polynomial.

Using a 3rd order Maclaurin polynomial, find an upper bound on the error when log(1+x) is approximated by a 3rd order polynomial for |x|<= 0.1. For some reason I keep getting a different answer ...
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1answer
37 views

Prove the taylor series of $ \cos(2z)$

First i turned $$\cos(2z) = \frac{e^{2iz} + e^{-2iz}}{2}$$, then using the taylor series of $$e^{z}$$I calculated the taylor series of both arguments. $$\frac{e^{2iz}}{2} = \sum_{n=0}^{\infty ...