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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Infinite (Taylor) Series

What is the general term for a Taylor series of $\sin(x)$ centered at $\pi/4$? It should be $(-1)^{[??]} \times \sqrt{2}/2 \times \frac{(x-\pi/4)^n}{n!}$ What power is $(-1)$ supposed to be raised ...
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1answer
668 views

Find the taylor series for $\cos x$ and indicate why it converges to $\cos x$ for all $x\in \Bbb R$.

Find the taylor series for $\cos x$ and indicate why it converges to $\cos x$ for all $x\in \Bbb R$. I've posted my own proof, I hope it is correct :)
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1answer
59 views

Exercise about MacLaurin's polynomial and small-o

In class the professor wrote the following limit: $\lim_{x\to 0} \frac{\sinh^2 (x) -x^2}{x^4}$ So he "expanded" (sorry for my English) the MacLaurin's formula for $\sinh x$ up to the 3rd power, and ...
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2answers
3k views

Solve the differential equation using Taylor-series expansion

Solve the differential equation using Taylor-series expansion: $$ \frac{dy}{dx} = x + y + xy \\ y (0) = 1 $$ to get value of $y$ at $x = 0.1$ and $x = 0.5$. Use terms through $x^5$.
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Showing that the Taylor series of $f(x)=\frac{1}{\sqrt{1+x}}$ converges to $f$ on (I think) the interval (-1,1].

Let $f(x)=\frac{1}{(1+x)^{1/2}}$. Supposed to find the Taylor series $Tf$ of $f$ around $0$, and show that it converges to $f$ on $[-1,1)$, (although I suspect there's a misprint in the book, and that ...
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1answer
1k views

First four Taylor Series Expansions

I'm supposed to write the first four taylor series expansions of $f(x=0)$ using: one term, two terms, three terms, four terms This is the function: $$f(x) = x^3 - 2x^2 + 2x - 3$$ Should I be using ...
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1answer
117 views

Taylor polynomial for $f(x)=x^{1/7}$ about $a=1$

Here is the problem: (a) Determine the Taylor polynomial $T_2(x)$ of degree $2$ for the function $f(x)=x^{1/7}$ centered at $a=1$. (b) Suppose we were to use the approximation $f(x) \approx ...
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1answer
304 views

Laurent series for an infinite sum of functions

I have an infinite sum of analytic functions that is guaranteed to converge for every $x$, except for $x=0$: \begin{equation} g(x) = \sum_{n=1}^\infty f_n (x) \end{equation} I want to expand the ...
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1answer
35 views

Radius of convergence of $(1+x)^p$

Problem: Show that $(1+x)^p$ converges everywhere for $p \in \mathbb{N}$, and for $|x| < 1$ otherwise. My work: I think that if $p \in \mathbb{N}$ then the Taylor series will just be a polynomial ...
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2answers
82 views

How to find a closed form for the derivatives of $F(x)=\frac1x\int_0^x\frac{1-\cos t}{t^2}\,dt,$ $F(0)=\frac12$?

I have been given the function $$F(x)=\frac1x\int_0^x\frac{1-\cos t}{t^2}\,{\rm d}t$$ for $x\ne 0,$ $F(0)=\frac12,$ and charged with finding a Taylor polynomial for $F(x)$ differing from $F$ by no ...
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137 views

On the hessian matrix and relative minima

I'm asked to prove the following statement: Let $\mathbb{A} \subseteq \mathbb{R}^n$ an open subset and $f: \mathbb{A} \to \mathbb{R} / f \in \mathbb{C}^3 \wedge (P \in \mathbb{A} : \nabla f(P)=0)$. ...
3
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2answers
297 views

Express $\sin nx$ and $\cos nx$ in terms of $\sin x$ and $\cos x$ respectively

What are the expansions of $\sin nx $ and $\cos nx$ in terms of $\sin x$ and $\cos x$ respectively? (here $n \in \mathbb N$). Maybe this is solved problem or there is new technique to ...
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1answer
53 views

Taylor series- help! [closed]

$f(x)$ is twice differentiable function in $[0,1]$. we know that: $f(0)=0$, $f(1)=1$, $f'(0)=f'(1)=0$. show that there exists a point $c$ such that $\left|f''(c)\right|\ge 4$ Thanks in advance
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1answer
310 views

Proof of the correctness of Taylor series

I am looking at the proof provided on the wiki page for taylor series http://en.wikipedia.org/wiki/Taylor%27s_theorem#Proof_for_Taylor.27s_theorem_in_one_real_variable One of the proof provided is ...
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2answers
63 views

Please help me understand Rudin Theorem 5.15

I am having trouble understanding the intuition behind the last part of this theorem. I'd appreciate some help understanding the intuition behind the last equation: $f(\beta ) = P (\beta ) + ...
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1answer
90 views

show that $\lim_{x\to \frac14} \frac{d^{2k-1}}{dx^{2k-1}}\cot(\pi x)=-(2\pi)^{2k-1}2^{2k}(2^{2k}-1) \frac{\left | B_{2k} \right |}{2k} $

I try to prove the relation between Polygamma function and Bernoulli numbers but I faced this problem,is how to show that $$\lim_{x\to \frac14} \frac{d^{2k-1}}{dx^{2k-1}}\cot(\pi ...
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1answer
109 views

When computing the Taylor series of $(\cos x)^2$ how does the slide jump to concluding it is $1-(\sin x)^2$?

In the following slide it shows how the taylor series of $(\cos x)^2$ is computed: On the first line they simply take the taylor series of cosx and write it out twice, which makes sense. However, ...
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113 views

Determine the Maclaurin series expansion of $\frac{\mathrm{exp}(z)}{(z+1)}$

Determine the Maclaurin series expansion of $\frac{\mathrm{exp}(z)}{(z+1)}$. This is the composition of the series expansion of the exponential function centered about $z = -1$. We can rectify the ...
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2answers
187 views

Taylor Expansion of the 1/2th Derivative

In trying to solve the problem $\sqrt D f(x)=g(x)$ I tried to expand the derivative as a Taylor series, and have encountered a lot of problems. Is there some reason that this shouldn't be possible? ...
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1answer
71 views

What's incorrect with this Taylor series derivation?

Let's \begin{align} f(T)= &f(0)+ \int_0^T f' (t)dt\\ f(T)=&(0)+f' (T)T-\int_0^T f'' (t)tdt\\ f(T)=&(0)+f' (T)t-f'' (T) \frac{T^2}{2}+\int_0^Tf''' (t) \frac{t^2}{2} dt\\ f(T)=&f(0)+f' ...
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102 views

Behavior of differential equation as argument goes to zero

I'm trying to solve a coupled set of ODEs, but before attempting the full numerical solution, I would like to get an idea of what the solution looks like around the origin. The equation at hand is: ...
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1answer
40 views

Taylor polynomial and degree

I read that one can form Taylor polynomials for some functions, like $$\sin x\approx x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!}.$$ Is it correct to say that $\sin x$ has no Taylor polynomial ...
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1answer
4k views

First Four Nonzero Terms of Taylor Series

I'm not acing Calculus II and while this problem might be easy to some, it's not to me. Any help will do. Problem: Find the first nonzero terms of the Taylor Series for $f(x) = \cos x$ where $$a ...
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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
58 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
74 views

Approximating $\sqrt{101}$ using Taylor series methods

I'm trying to approximate $\sqrt{101}$ using the Taylor series for the function $f(x)=\sqrt{x}$ centered at the point $x=100$. I need to obtain an approximation that is within $0.01$ of the correct ...
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1answer
82 views

Taylor expansion of a not easily differentiable function

Context: I'm trying to find the period of a simple pendulum. As is well known, if the initial angle is small the period is approximately constant. I'm trying to do a second order expansion. I have ...
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1answer
24 views

Help with Taylor Series

I am trying to find a Taylor series for the following function: ${1\over 1-9x}$ centered at c = 7 I browsed through my Calc II book and found that I can use the general formula for a Taylor series ...
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2answers
78 views

Decide convergence of the series

Using Taylor expansion decide convergence of the series: $$\sum_{n=1}^{\infty}(e-(1+{{1}\over{n}})^n)^p = \sum_{n=1}^{\infty}a_n$$ I expanded $a_n$ like this $a_n = (e-(1+{{1}\over{n}})^n)^p = ...
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1answer
82 views

Finding an asymptotic expansion for a transcedental equation

I am new around here and was hoping you will be able to help me with the following. I have the equation: $x^3 - 3x^2 +(3-\epsilon ) x + \epsilon = sin(\frac{\pi}{2} x +\frac{\pi \epsilon}{2} ) $ and ...
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1answer
33 views

Finding $\frac{\partial ^8 f}{\partial x^4\partial y^4}$

Given the function $f(x,y)=\frac{1}{1-xy}$ find the value of$\frac{\partial ^8 f}{\partial x^4\partial y^4}(0,0)$. First I developed the function into a taylor series using geometric series ...
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98 views

If it converges, how to show that power series converges to $f(x)$?

I had a very basic question. Suppose $f(x)$ is a function. And let us say it has a power series :- $$f(x) = \sum_{n=0}^\infty a_nx^n.$$ Suppose we are operating inside the region of convergence. ...
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1answer
66 views

Showing $|a_k | \le 1$

Let $A$ be the closed unit disk $A= \{z \in \mathbb{C}: |z| \le 1 \}$. Suppose $f$ is an entire function whose Taylor series centered at the origin is $$\sum_{k=0}^{\infty} a_kz^k$$ and that $f$ maps ...
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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) + ...
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1answer
64 views

Mistake in Taylor expansion?

Given: The first derivative of $\tan x$ is $1/\cos^2 x$ So the derivative of $\tan x$ when $x=0$ should be $1$. This derivative times $x$ should be a term in the Taylor expansion (the term then being ...
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1answer
125 views

Convergence radius of $\sqrt{\cos(z)}$

Compute the first 3 non zero terms of the Taylor expansion of $\sqrt{\cos(z)}$ at $z=0$ and determine its convergence radius, considering only the principal branch of the square root. I've computed ...
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1answer
174 views

Why do we use big Oh in taylor series?

In the taylor series for sin(x), we write: $$ \sin{x} = x + \frac{x^3}{6} + \frac{x^5}{120} + O(x^7) $$ Meaning that $\sin{x} = x + \frac{x^3}{6} + \frac{x^5}{120}$ and terms of order $x^7$ and ...
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Prove Taylor expansion with mean value theorem

On http://vapour-trail.blogspot.com/2006/03/brief-explanation-of-taylor-series-via.html one can find an hint at how to derive Taylor expansions from the mean value theorem. The process goes as ...
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1answer
102 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 ...
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1answer
89 views

Intuition behind 2nd order approximation, help please

I know how to apply the formula for Taylor Expansions. But what I want to understand is the intuition. Let me explain with the following example: If $y=x^5$ its 1st derivative is $5x^4$ and its 2nd ...
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1answer
495 views

Taylor series with functions as parameters (as opposed to variables)

I'm doing my own research on the Euler-Lagrange equation and came across a proof in van Brunt's textbook "The Calculus of Variations". However, there is something I don't quite understand. Here is an ...
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154 views

How do I obtain the Laurent series for $f(z)=\frac 1{\cos(z^4)-1}$ about $0$?

I know that $$\cos(z^4)-1=-\frac{z^8}{2!}+\frac{z^{16}}{4!}+...$$ but how do I take the reciprocal of this series (please do not use little-o notation)? Or are there better methods to obtain the ...
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64 views

Prove that $\cos x=1-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}+R_8(x)$ where $|R_8(x)|\leq \frac{x^8}{8!}$

Prove that $$\cos x=1-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}+R_8(x)$$ where $|R_8(x)|\leq \frac{x^8}{8!}$
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928 views

Proof of the “Radius of Convergence Theorem”

I can't figure out how it is valid to invoke the Absolute Convergence Theorem, whose hypothesis is "Let the power series have radius of convergence R", to establish case c of the Radius of Convergence ...
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1answer
295 views

Taylor expansion of an integral

I am interested in the Taylor series expansion around $t=0$ of the following expression: $$I(t)=\int_{0}^{\infty}e^{-x^2}\log\left(e^{-(x-t)^2}+e^{-(x+t)^2}\right)dx$$ Normally, I would proceed by ...
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1answer
174 views

Complex Taylor series and Bernoulli numbers

Let: $f(z)=\frac{z}{e^z-1}$ if $z\ne0$, and $f(z)=1$ if $z=0$. Please help to prove that $\sum_{k=0}^{n-1}\binom n kf^{(k)}(0)=0$ for any $n>1$ and $f^{(2n+1)}(0)=0$ for any natrual $n$.
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1answer
208 views

Orthogonal complete set of functions

Every square-integrable function on an interval can be written as a linear combination of e^inx (Fourier series). Are there any other orthogonal and complete set of functions for square integrable ...
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1answer
98 views

Taylor expansion of $H = \sqrt{m^2 - \hbar^2 \nabla^2}$

$$ H = \sqrt{m^2 - \hbar^2 \nabla^2} $$ Suppose that there is a equation like this. How do you taylor-expand this equation? I am extremely confused.
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1answer
93 views

Repeated integration of a real valued bounded function.

I was reviewing the proof the remainder estimate for a Taylor series expansion and I came across something I can't find an intuitive explanation for: if you have a function f that's bounded on an ...
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1answer
865 views

Difference in limits because of greatest-integer function

A Problem : \begin{equation}\lim_{x\to 0} \frac{\sin x}{x}\end{equation} results in the solution : 1 But the same function enclosed in a greatest integer function results in a 0 ...