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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Estimate the degree of a Taylor Polynomial using its Error Term

In my 2nd year studying Maths at Uni and revising for a Numerical Analysis final exam. We're given 1 past paper but no solutions, and I can't answer this question: Use the error term of a Taylor ...
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1answer
49 views

why start the taylor series of $\cos^{2} x$ at $k=1$ and not just $k=0$ as I do not understand the problem with $2^{-1}$

Im using $\cos^2 x=\frac{1}{2}(1+\cos(2x))$ and $\cos x = (-1)^k \frac{(2x)^{2k}}{(2k)!}$ to find the sum for the Taylor series of $\cos^2 x$. I thought I was getting it. When I find the answer ...
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3answers
73 views

Expand function into a Maclaurin's series

The function is given with: $f(x)=\dfrac{x^{2012}}{(1-x^3)^2}$ I have no idea what to do here and honestly, I don't really understand what a Maclaurin's series is. I know the definition but I don't ...
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2answers
90 views

Why can't you find all antiderivatives by integrating a power series?

if $f(x) = \sum\limits^{\infty}_{n=0}\frac{f^{(n)}(0)}{n!}x^n$ why can't you do the following to find a general solution $F(x) \equiv \int f(x)dx$ $F(x) = \int ...
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1answer
143 views

Question about a solution to a problem involving Taylor's theorem and local minimum

I've been studying "Berkeley Problems in Mathematics, Souza, Silva" and I came across this problem: Let $f : \mathbb{R} \rightarrow \mathbb{R}$ be a $C^{\infty}$ function. Assume that $f(x)$ has a ...
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1answer
235 views

Maclaurin series of $f(x)=\sinh(1/x)$?

As we know the formula of Maclaurin series for $f(x) = \sinh(x)$ is $f(x)=x+x^3/3! + x^5/5!+\ldots$ Could anyone tell me what is the Maclaurin series of $f(x)=\sinh(1/x)$?
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3answers
642 views

Trigonometric identities using $\sin x$ and $\cos x$ definition as infinite series

Can someone show the way to proof that $$\cos(x+y) = \cos x\cdot\cos y - \sin x\cdot\sin y$$ and $$\cos^2x+\sin^2 x = 1$$ using the definition of $\sin x$ and $\cos x$ with infinite series. thanks...
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1answer
734 views

Maclaurin series for sin(x) representation

The Maclaurin series for $\sin(x)$ is: $$ \sin(x) = x - \dfrac{x^3}{3!} + \dfrac{x^5}{5!} - \dfrac{x^7}{7!} ... $$ Which according to wikipedia is: $$ \displaystyle \sum_{n=0}^{\infty} ...
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1answer
115 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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2answers
52 views

Other log solutions?

I am evaluating the expression: $\ln(1)$ And I know the trivial solution is $0$. Are there other solutions to this equation? I feel there should be, my logic is as follows: if: $\ln(1) = x ...
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1answer
121 views

Integral remainder converges to 0

I want to show that $\displaystyle \log(1+x)=\sum_{n=1}^{\infty}(-1)^{n-1}\frac{x^n}{n}$ for $-1<x\leq1$ i want to show it with the integral remainder of the taylor series that gave me: ...
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2answers
48 views

Finding the sum of a Taylor expansion

I want to find the following sum: $$ \sum\limits_{k=0}^\infty (-1)^k \frac{(\ln{4})^k}{k!} $$ I decided to substitute $x = \ln{4}$: $$ \sum\limits_{k=0}^\infty (-1)^k \frac{x^k}{k!} $$ The first ...
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2answers
2k views

Tangent Line Error Bound with Taylor Series

I have an equation, $e^x$, based at 0 (b=0). I am supposed to us the Tangent Line Error Bound to bound the error $|f(x)-T1(x)|$ on the interval I=[-1,1]. (Aside: I have already computed the first ...
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80 views

optimal control -Taylor expansion - PDE problem

I am trying to follow perturbation analysis in this paper (Optimal control of fluid limits of queuing networks and stochasticity corrections) and I am stuck at one point. For the given control ...
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1answer
34 views

Finding power series for $f(x) = \frac{4x+53}{x^2-x-30}$

Given $f(x) = \dfrac{4x+53}{x^2-x-30}$, display it as a power series and find the radius of convergence. then calculate $f^{(20)}(0)$ So what I did was look at the Taylor Series Formula: $$f(x) = ...
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2answers
93 views

Taylor series of $f(x)=\frac {e^x-1}{x}$

I am asked to expand $f(x)=\frac {e^x-1}{x}$ centered at 0 using the known Talyor series of functions. How to simplify the function so that it can be expanded more easily?
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1answer
59 views

Is $e^z\sum_{k=0}^\infty\frac{k^3}{3^k}z^k$ analytic inside $|z|=3$?

Am I correct that the following function is analytic at least inside $|z|=3$? (I used the ratio test.) The solutions manual says that the function is analytic on and inside |z|=1, so I wonder if I'm ...
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0answers
167 views

Maclaurin series expansion of an expression that involves a fraction

In the context of statistical mechanics the "classical trace" is defined as $Tr(A e^{-\beta H}) = \int dr^N dp^N A e^{-\beta H}$ where $r^N$ and $p^N$ are phase space variables. So if $\Delta H$ is a ...
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1answer
175 views

An issue with approximations of a recurrence sequence

By trying to give an approximation to a given recurrence sequence I encountered a problem. To be more precise I have a method but it fails if the right condition is not met and I wonder how I should ...
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1answer
382 views

taylor expansion of exponential function

To prove CLT of binomial distribution, $$X \sim \mbox{bin}(n,p)$$ $M_X(t)=(p e^t+q)^n$ where $M$ is mgf. Let $Z=\frac{X-np}{ \sqrt{npq}}$, $\sigma =\sqrt{npq}$, then $$ \begin{align} ...
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3answers
215 views

Finding the Taylor series of $f(z)=\frac{1}{z}$ around $z=z_{0}$

I was asked the following (homework) question: For each (different but constant) $z_{0}\in G:=\{z\in\mathbb{C}:\, z\neq0$} find a power series $\sum_{n=0}^{\infty}a_{n}(z-z_{0})^{n}$ whose sum ...
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2answers
100 views

Series expansion with remaining $log n$

I'm studying the asymptotic behavior $(n \rightarrow \infty)$ of the following formula, where $k$ is a given constant. $$ \frac{1}{n^{k(k+1)/(2n)}(2kn−k(1+k) \ln n)^2}$$ I'm trying to do a series ...
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1answer
438 views

Taylor series with function composition

Pretty simple, but I want to take the first order taylor series expansion of the following: $f(g(x,y+Δy))$ Would the following be correct? $f(g(x,y+Δy)) = f(g(x,y) + \frac{\partial}{\partial ...
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3answers
74 views

Taylor Polynomial for $x^{1/3}$

a. Compute the Taylor polynomial $T_3(x)$ for the function $(x)^{1/3}$ around the point $x=1$. b. Compute an error bound for the above approximation at $x = 1.3$. I'm having trouble figuring ...
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123 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 $x_0\epsilon (0,x)$ ...
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2answers
3k views

Multiplying Taylor series and composition

I have two questions: A. I know the taylor series for $\arctan(x)$ and for $e^x$. How do I find the series for $\arctan(x)\cdot e^x$ ? B. Say I want to find the series for $\arctan(g(x))$, do I just ...
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134 views

Taylor series of a vector field?

Can someone give me the definition of the Taylor Series of a vector field in $\mathbb{C}^2$? Thanks!
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1answer
67 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 ...
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1answer
180 views

Finding the error of the Taylor expansion of $\log(1 + x)$

The questions is as defined below. Let $f(x)= \log(1+x)$. Show that the Taylor remainder $R_{0,k}(x)$, defined by $$R_{a,k}(x)= f(a+x) - P_{a,k}(x) = f(a+h) -\sum_{j=0}^{k} \frac ...
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65 views

Binomial Expansion problem error

I tried solving this question but failed. a) Expand $(1+2x)^{1/4}$ in ascending powers of $x$ up to and including the term in $x^3$, simplifying each term as far as possible. b) By substituting ...
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1answer
116 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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1answer
315 views

$x^3-3x-3=0$, prove that $10^x<127$

$x$ is the real root of the equation $$3x^3-5x+8=0,\tag 1$$ prove that $$e^x>\frac{40}{237}.$$ I find this inequality in a very accidental way,I think it's very difficult,because the actual value ...
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167 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 ...
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2answers
92 views

How does one get the Bernoulli numbers via the generating function?

Here is the definition: Bernoulli numbers arise in Taylor series in the expansion $$\frac{x}{e^x-1}=\sum_{k=0}^\infty B_k \frac{x^k}{k!}$$ I've tried to naively expand $\frac{x}{e^x-1}$ around ...
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1answer
107 views

Itō's Lemma neglecting terms

In my project I am trying to give a Heuristic proof of Itō's lemma. I show $E[dW_t^2] = dt$ I take $g(x,t)$ to be a twice continuously differentiable function and $dt$ to be infinitesimally small. ...
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2answers
323 views

taylor expansion of an integral $\int_0^1{e^{x^2}}$

I need to calculate $\int_0^1{e^{x^2}\:dx}$ with taylor expasin in accurancy of less than 0.001. The taylor expansion around $x_0=0$ is $e^{x^2}=1+x^2+\frac{x^4}{3!}+...$. I need to calculate when the ...
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1answer
272 views

Prove that d/dx (sin x) = cos x, using Taylor series

Show by differentiation of the series for sin x that $$\frac{d}{dx} (\sin x) = \cos x$$ (Using Taylor series.) If you can given an indication or solved answer to my question would be great. Thanks ...
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79 views

Determine the series for cos x^2

Use the series for Cos x (Taylor Series) If you could give me help or the solution to the problem, that would be great! Thanks
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1answer
47 views

Points around which one expands and the radiuses of convergence

I'm trying to make sense of the following passage: Let $f(x)=\frac{1}{x+1}$ and $R_0$ the radius of convergence of the Taylor series of $f$ around $x_0=0$, analogously: $R_1$ — around ...
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528 views

Taylor Series for $e^x$ where $x = 1$, estimating the Error

I'm trying to calculate $e$ to a certain number of digits. The Maclaurin Series expansion of $\displaystyle e^x = \sum_{n=0}^\infty \frac{x^n}{n!}$. When $x = 1$ we can approximate the value of $e$ by ...
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2answers
82 views

Multiplication of two Taylor expansions

I'm trying to calculate a Taylor expansion which is : $\cos(x). exp(x)$ in the neighborhood of 0 in order 3 this is the result I got : $$\cos(x). exp(x) = ...
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490 views

In Taylor series, what's the significance of choosing the point of expansion $x=a$?

So I read about the Taylor series and it said you can choose to expand the series around a given point ($x=a$). Does it matter which point you choose in calculating the value of the series? For ...
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2answers
167 views

A question on the convergence of a Taylor series of some prominent function

The function $f:\mathbb{R}\to\mathbb{R}$ defined by $$ f(x)=\begin{cases}e^{-\frac{1}{x^2}} &if &x\neq 0\\ 0 & else \end{cases} $$ is a prominent example of a function whose Taylor series ...
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1answer
78 views

Wave-Function Series?

So I was basically exploring the function: $\displaystyle {\text{frac}(x)}$ which is the fractional part function and I noticed that it has a nice fourier series definition which is: ...
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149 views

Taylor series expansion example

I was reading an article and there was a snippet with a taylor series expansion as shown below: My question is, should (11) read as $F(xA+h)+(xΔA+Δh)\frac{\partial}{\partial x}F(xA+h)$ instead of ...
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391 views

Taylor Series Expansion with e and sin

Show that when $z\neq0$, (a) $$\frac{e^z}{z^2}=\frac{1}{z^2}+\frac{1}{z}+\frac{1}{2!}+\frac{z}{3!}+\frac{z^2}{4!}+...$$ (b) ...
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1answer
75 views

Maclauren Series and taylor polynomials

Question: Suppose that the function $k(x)$ has a maclauren series that converges $\left(-\frac{1}{2} , \frac{1}{2}\right]$ and you are told that $|k^{(n)}(x)| \leq 10$ at all $|x| \leq ...
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3answers
180 views

Summation of a series.

I encountered this problem in Physics before i knew about a thing called Taylor Polynomials My problem was that i had to sum this series : ...
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1answer
127 views

Taylor polynomial approximation

How do you determine if adding more terms to the Taylor polynomial will improve its approximation of $f(p)$ or in other words, how do you determine if a Taylor series converges for a particular value ...
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1answer
333 views

Using the series of $\tan^{-1}(x)$ for calculating $\pi$

The power series expansion of $\tan^{-1}(x)$ is $$\tan^{-1}(x)=x-\frac 13 x^3+\frac 15 x^5-\frac 17 x^7+ \cdots .$$ Use the above series to determine a series for calculating $\pi$.