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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Taylor Series of $2xe^x$

I have to find the Taylor Series for $2xe^x$ centred at $x=1$. I came up with the following. $$e^x = e^{x-1} \times e = e \bigg( \sum_{n=0}^\infty \frac{(x-1)^n}{n!}\bigg)$$ Then consider $2xe^x$. ...
4
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3answers
197 views

Taylor series: $\sin x = x$?

Taylor series are used to expand a function to a series of functions that sometimes makes calculations easier. The more terms of a series we consider the more precise the solution would be. ...
4
votes
2answers
1k 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 ...
4
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2answers
341 views

Taylor series Question

So I have a test next week and I saw this question with no answer and I would like to some help. Let $f: \mathbb{R} \rightarrow \mathbb{R}$ infinitely differentiable and let $\sum _{n=0}^{\infty} ...
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1answer
6k views

Difference between the Laurent and Taylor Series.

I have never taken complex analysis, but I am preparing for a GRE for this week end and I am trying to learn a bit about Laurent Series. So far what I get is that the Laurent Series are of form ...
4
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3answers
281 views

Rewrite trigonometric expression to be be numerically “stable”

Is it possible to write the following function: $$ f(x) = \begin{cases} \frac{x-\sin x}{1- \cos x}& x\neq 0\\ 0 & x=0 \end{cases} $$ as a composition of elementary functions (including ...
4
votes
3answers
510 views

Calculate $\sqrt{5}$ using taylor series to the accuracy of 3 digit after the point $(0.5*10^{-3})$.

I need to calculate $\sqrt{5}$ using taylor series to the accuracy of 3 digit after the point $(0.5*10^{-3})$. I defined : $$f(x)=\sqrt{x}$$ Therefore : $$f'(x)=\frac{1}{2\sqrt{x}}$$ ...
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2answers
211 views

Analogy to the purpose of Taylor series

I want to know an analogy to the purpose of Taylor series. I did a google search for web and videos : all talks about what Taylor series and examples of it. But no analogies. I am not a math geek and ...
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2answers
29k views

taylor series of ln(1+x)?

Compute the taylor series of $ln(1+x)$ I've first computed derivatives (upto the 4th) of ln(1+x) $f^{'}(x)$ = $\frac{1}{1+x}$ $f^{''}(x) = \frac{-1}{(1+x)^2}$ $f^{'''}(x) = \frac{2}{(1+x)^3}$ ...
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2answers
88 views

Is $\cos(\frac{\pi}{3})$ exactly equal to 0.5 or approximately equal to 0.5

We know that $\cos(\frac{\pi}{3})=\frac{1}{2}$, but the expansion for $\cos(x)$ is as follows: $$ \cos(x)=1-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}+\ldots$$ So this would make $$\begin{align} ...
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2answers
102 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 ...
4
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1answer
2k views

Find Maclaurin series of $(\sin(x^3))^{1/3}$

How do I find Maclaurin series for the function: $$\sqrt[3]{\sin(x^3)}$$ The answer should be: $$ x - \frac {x^7}{18} - \frac {{x}^{13}}{3240} + o(x^{13})$$ I tried: $$\sin x = x - \frac ...
4
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1answer
679 views

Is the Taylor series comparable to Fourier series and spherical harmonics?

I am currently trying to grasp spherical harmonics and try to digest that we proved that the sine and cosine functions are a basis for the $L^2$ space of the squared-integrable functions. So as far ...
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2answers
8k 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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2answers
1k views

How do we know Taylor's Series works with complex numbers?

Euler famously used the Taylor's Series of $\exp$: $$\exp (x)=\sum_{n=0}^\infty \frac{x^n}{n!}$$ and made the substitution $x=i\theta$ to find $$\exp(i\theta) = \cos (\theta)+i\sin (\theta)$$ How ...
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1answer
281 views

Extending partial sums of the Taylor series of $e^x$ to a smooth function on $\mathbb{R}^2$?

Is there a smooth function $f:\mathbb{R}^2 \to \mathbb{R}$ such that $f(x,n)$, where $n\in\mathbb{N}$, is the truncated Taylor series of $e^x$, namely $1+ x + \frac{x^2}{2} + \dotsb + \frac{x^n}{n!}$, ...
4
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3answers
111 views

computing the series $\sum_{n=1}^\infty \frac{1}{n^2 2^n}$

$$\sum_{n=1}^\infty \frac{1}{n^2 2^n}$$ I am new in series thus I tried a pair of methods to compute but I couldn't
4
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1answer
84 views

Limitations of fractional derivative approximation with Taylor series

I was playing around with the concept of fraction derivatives, and came across some base functions for which it is defined, namely power and exponential functions $$ \left(\frac{d}{dt}\right)^\alpha ...
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2answers
121 views

Maclaurin series expansion for $e^{-1/x^2}$

I am currently extremely confused on how to proceed with the Maclaurin series expansion for my current function. I got my derivatives and I got my formula, however, plugging them in gives me a ...
4
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2answers
97 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 $ ...
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3answers
263 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 ...
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2answers
1k views

Taylor polynomial of $\int_{0}^{x}\sin(t^2)dt$

I just learned about Taylor polynomials, and I am trying to estimate $\int_{0}^{1/2}\sin(x^2)dx$ using the 3rd degree Taylor polynomial of $F(x)=\int_{0}^{x}\sin(t^2)dt$ at $0$. I get the following: ...
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3answers
75 views

Expand in Taylor series $\frac{1}{1-\sin{x}}$

Expand in Taylor series $\frac{1}{1-\sin{x}}$ I have an idea that $\frac{1}{1-\sin{x}} = 1 + \sin {x} + \sin^2 {x} + \sin^3 {x} + \dots$ But I don't know what to do next. Every sine expands in ...
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2answers
4k views

Third order term in Taylor Series

What is the third order term in the Taylor Series Expansion? I know it will just be third order partial derivatives but I want to know how is it expressed in a compact Matrix notation. For instance ...
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2answers
1k views

Taylor polynomial about the origin

Find the 3rd degree Taylor polynomial about the origin of $$f(x,y)=\sin (x)\ln(1+y)$$ So I used this formula to calculate it ...
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1answer
715 views

Is exponential function analytic over all complex numbers

In my textbook, I find a text where it says $e^z$ is analytic everywhere (in complex plane). Is it true? If so, what is the proof? I approached using maclaurin series, which gives $e^z= ...
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2answers
2k 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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4answers
2k views

What exactly are the “higher order terms” (H.O.T.) in Taylor series expansion in multivariable case?

I know the Taylor series expansion in single variable case: $$ f(x) = f(x_0) + f'(x_0)(x - x_0) + \frac{1}{2}f''(x_0)(x-x_0)^2 + \frac{1}{3!}f^{(3)}(x_0)(x-x_0)^3 + \frac{1}{4!}f^{(4)}(x_0)(x-x_0)^4 ...
4
votes
6answers
144 views

$\lim_{x \to 0} \frac {(x^2-\sin x^2) }{ (e^ {x^2}+ e^ {-x^2} -2)} $ solution?

I recently took an math exam where I had this limit to solve $$ \lim_{x \to 0} \frac {(x^2-\sin x^2) }{ (e^ {x^2}+ e^ {-x^2} -2)} $$ and I tought I did it right, since I proceeded like this: 1st I ...
4
votes
2answers
124 views

$f(y) \leq f(x)+\nabla f(x)\cdot (y-x) $ and $f(x)\geq 0$ implies that $f$ is constant.

Here is the question. Suppose that $f: \mathbb R^n \rightarrow \mathbb R$ has two derivatives and the associated hessian matrix is negative semidefinite on all of $\mathbb R^n$. Show that for any ...
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3answers
1k views

Taylor Series of Ratio of Bessel Functions

In attempting to solve a recursion relation I have used a generating function method. This resulted in a differential equation to which I have the solution, and now I need to calculate the Taylor ...
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2answers
620 views

Taylor polynomial with Lagrange remainder

In my course there's a paragraph: Taylor polynomial with Lagrange remainder, The paragraph starts with a theorem (I left out the constraints): $$ ( \exists \theta \in ]0,1[)(f(a +h) = T_{f,a,n}(a + ...
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2answers
120 views

Infinite Sum without using $\sin\pi$

What's a purely algebraic way to prove that $\pi-\frac{\pi^3}{3!}+\frac{\pi^5}{5!}-\dots=0$? I'm sure that the first step is to write $\pi=4-\frac43+\frac45-\dots$, but I haven't been bold enough to ...
4
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2answers
203 views

Applying Taylor theorem on a linear map

I found the following in a stack of practice problems but had trouble dealing with it: Consider a linear map $A:C^\infty(\mathbb{R}^n)\rightarrow \mathbb{R}$ such that: If $f\in ...
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3answers
100 views

An Expression of $\ln(2)$

I saw online that the following infinite series has a value of $\ln(2)$: $\sum_{n=0}^\infty \left(\dfrac{1}{n+1}-\dfrac{1}{n+2} +\dfrac{1}{n+3}-\cdot\cdot\cdot\right)^2$ I started off by defining ...
4
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3answers
61 views

Why, intuitively, does the Maclaurin series for $e^x$ but not $\ln(1+x)$ converge globally?

So we all know that, $\forall x\in\mathbb{R}$, $$e^x = \sum_{k=0}^{\infty}\frac{x^k}{k!}$$ And that $$\ln (1+x) = \sum_{k=0}^{\infty} \frac{(-1)^{k-1}}{k}x^k$$ But that this only holds for ...
4
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1answer
39 views

Second order differential

Suppose I have a function $f = f(x,y,z)$. Then, the first order differential, or the linear approximation is $$ df = \frac{\partial f}{\partial x}dx + \frac{\partial f}{\partial y}dy + \frac{\partial ...
4
votes
2answers
68 views

Limiting value of $\frac{x^n e^x}{n!}$ as $n\to\infty$

For the Taylor Series the remainder is of the form $$R_n = \frac{(x-a)^n}{n!} f^{(n)}(\xi) $$ with $a \leq \xi \leq x$ For the series of $e^x$ about $0$ (that is, the Maclaurin series) the remainder ...
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1answer
260 views

Taylor series approximation of function under norm

I am reading this paper. At page number $4$, term $||Au - f||^{2}$ is approximated by taylor series approximation around $u^{k}$. The resulting approximations are $$\|Au - f\|^{2} \approx ...
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1answer
93 views

Expansion of $f(x)=\frac {\ln(1+\sin ax)-x(1+\arctan x)^{1/x}}{1-\cos x} $ for $x$ near $0$

How to find finite expansion of $$f(x)=\frac {\ln(1+\sin(ax))-x(1+\arctan x)^{1/x}}{1-\cos x} $$ to order $2$ and neighborhood $0$. My Dr. didn't expand them all to order $2$, please I have problem ...
4
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1answer
76 views

estimation of $\pi$ and $e$ by using the Taylor series of $\cos x$

how can one show, that $3<\pi<3.2$, $2.7<e<3$ by just knowing, the estimation of $\cos(x)$, namely: $$1-x^2/2+x^4/24-x^6/720\le\cos(x)$$ ? If I substitute Pi/2 into this estimation, I ...
4
votes
1answer
28 views

Equality involving Taylor coefficients

Considering the following series expansion $$ \frac{1}{{1 - 2x - x^2 }} = \sum\limits_{n = 0}^\infty {a_n } x^n $$ prove that $ \forall n,\,\exists m $ such that $ a_n ^2 + a_{n + 1} ^2 = a_m ...
4
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1answer
174 views

Unbiased asymptotic variance

Problem: Let $X_1,...,X_n$ be indep. r.v.'s that satisfy, for $i = 1,...,n$, $E(X_i) = \mu_i(\theta)$ & $\mathrm{Var}(X_i)= \sigma_i^2(\theta)$. $\theta$ is the parameter of interest and the ...
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2answers
169 views

Intuition behind Taylor/Maclaurin Series

** This is a different question than Intuition explanation of taylor expansion? ** I understand some of the intuition behind a Taylor/Maclaurin expansion. More specifically, I understand that adding ...
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2answers
125 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 ...
4
votes
3answers
186 views

What is the relationship between saying “a Taylor series converges for all $x$” and “a Taylor series converges to a function, f(x)”

Given the following Taylor series: $1-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}+\frac{x^8}{8!}- \dots$ We know that: It converges for all of $x$ It converges to the function $\cos x$ The ...
4
votes
2answers
587 views

Let $g(x)=e^{-1/x^2}$ for $x\not=0$ and $g(0)=0$. Show that $g^{(n)}(0)=0$ for all $n\in\Bbb N$.

Let $g(x)=e^{-1/x^2}$ for $x\not=0$ and $g(0)=0$. Show that $g^{(n)}(0)=0$ for all $n\in\Bbb N$. In the text it is already proven that for the function $f$ with $f(x)=e^{-\frac{1}{x}}$ for $x>0$ ...
4
votes
2answers
100 views

counting the number of real zeros, and find limitation

Let$$P_n(x)=\sum_{k=0}^{n}(-1)^k\frac{x^{2k+1}}{(2k+1)!}$$ let $c_n$ be the number of real zeros of $P_n$. determine$$\lim_{n \rightarrow \infty}\frac{c_n}{2n+1}$$
4
votes
1answer
80 views

How fast does this grow: $f(x) =\sum\limits_{i=1}^{\infty} \dfrac{x^i}{i!^{1/2}}$ for real $x$?

How fast does this grow: $$f(x) =\sum_{i=1}^{\infty} \dfrac{x^i}{i!^{1/2}}, \qquad x\in\mathbb{R}?$$ Is it faster than $e^{x^2}$? If Im not mistaken the Bell numbers grow fast enough such that ...
4
votes
2answers
48 views

How to compute $\lim_{x \to 0} (\frac{x^5 e^{-1/x^2}+x/2 - \sin(x/2))}{x^3})$?

I have a problem with this limit. I have no idea where is the problem. Can you correct my mistake? Thanks $$\lim\limits_{x \to 0} \left(\frac{x^5 e^\frac{-1}{x^2}+\frac{x}{2} - ...