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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A Taylor Expansion of a Stochastic Process

As part of a binomial model of a stochastic process, my professor claims that the Taylor Expansion of: $$x\pm = 1 \pm (e^{\sigma^{2}h} - 1)^{1/2}$$ is: $$x = 1 \pm \sigma \sqrt h + O(h^{3/2}) $$ ...
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1answer
33 views

How would you use Maclaurin Series in this question? [closed]

How would you solve $\lim_{x\to0} \frac{1-\cos(x)}{x^2}$ using MacLaurin series?
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1answer
10 views

MacLaurin Series with 2 variables - error

I have a real function $f(x,y)$, where $x,y$ are real. For a fixed $x_{0}$ I want to expand $f(x_{0},y)$ in $y_{0}$ in a first order MacLaurin series How should I write the error with the big O ...
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2answers
66 views

Find a function for the infinite sum $\sum_{n=0}^\infty \frac{n}{n+1}x^n$

I need to find a function $f(x)$ which is equal to the sum $$ \sum_{n=0}^\infty \frac{n}{n+1}x^n, $$ for every $x\in \mathbb{R}$ for which the series converge. Now, using WolframAlpha, I've found the ...
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2answers
94 views

How to calculate the series $-\frac12-\frac14+\frac13-\frac16-\frac18+\frac15-\frac{1}{10}…$?

$-\frac12-\frac14+\frac13-\frac16-\frac18+\frac15-\frac{1}{10}...$ After rearrangement the series looks like $\sum^{\infty}_{n=2}\frac{(-1)^{n+1}}{n}$. My way of doing this is using Taylor series of ...
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1answer
29 views

Evaluating limits via Infinite Series

I am to evaluate the following limit of sums and quotients of infinite series $\lim\limits_{z \to 0} \frac{(z^3 + z^6 - z^9 + ...)+(2z^3 -2 z^5 + 2z^7 - 2z^9 ...)}{z^8 + z^{16} + x^{24} + ... }$. I ...
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1answer
11 views

Show that monotonicity implies positive definiteness of the Jacobian

Given $f: \mathbb{R}^n \to \mathbb{R}^n$, $f$ differentiable, $x,y, p \in \mathbb{R}^n$, show that $(x-y)^T(f(x) - f(y)) \geq 0 \Leftrightarrow p^TDf(x)p \geq 0, \forall p \in \mathbb{R}^n$ This ...
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0answers
16 views

If the derivative is written as shifts, can you relate it to the laplace/fourier tranform?

I was wondering if there is a way to write the derivative as an exponential? This might sound crazy at first, but I recently came across this formula for the Taylor expansion in three dimensions: ...
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3answers
87 views

How to evaluate this limit using Taylor expansions?

I am trying to evaluate this limit: $\lim_{x \to 0} \dfrac{(\sin x -x)(\cos x -1)}{x(e^x-1)^4}$ I know that I need to use Taylor expansions for $\sin x -x$, $\cos x -1$ and $e^x-1$. I also realise ...
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0answers
25 views

find a closed formula for: $\frac{1}{\sqrt{1-(\sin{t}\ \sin{x})^2}}$

I need to find a closed formula for: $\frac{1}{\sqrt{1-(\sin{t}\ \sin{x})^2}}$ By Taylor series in respect of $x$, but I can't find a pattern. Can anyone help me? I can use Wolfram alpha but it ...
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2answers
54 views

How to evaluate $\lim\limits_{x\to 0}\frac{e^{\arctan{(x)}}-xe^{\pi x}-1}{(\ln{(1+x)})^2}$?

How to evaluate $\lim\limits_{x\to 0}\frac{e^{\arctan{(x)}}-xe^{\pi x}-1}{(\ln{(1+x)})^2}$? So I think we expand to $x^2$ since the lowest term for $\ln(1+x)$ is $x$ Let $u=\arctan{(x)}$ ...
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0answers
44 views

Approximate roots of nonlinear equation (non-integer polynomial)

In case of pulsating bubble arising from underwater explosion, bubble radius satisfies the following equation. $x^3\dot{x}^{2} + x^3 + \frac{k}{x^{3(\gamma-1)}} = 1$ The minimum and maximum bubble ...
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1answer
39 views

Taylor Series for $f(x)$

A function $f$ is defined as $$ f(x) = \left \{\begin{aligned} &{cosx-1\over x^{2}} & for\,x \neq 0\\ &{-1\over 2} & for\,x=0 \end{aligned} \right. $$ Using the first three non zero ...
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2answers
21 views

How to compute $\sum^{\infty}_{n=2}\frac{(4n^2+8n+3)2^n}{n!}$?

How to compute $\sum^{\infty}_{n=2}\frac{(4n^2+8n+3)2^n}{n!}$? I am trying to connect the series to $e^x$ My try: ...
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1answer
10 views

How to compute taylor series of $\sin{x}$ about $a=\frac{\pi}{4}$?

How to compute taylor series of $\sin{x}$ about $a=\frac{\pi}{4}$? We know $\sin{x}=\sum^{\infty}_{n=0}\frac{(-1)^nx^{2n+1}}{(2n+1)!}$. Let $t=x-\frac{\pi}{4}$, then $t+\frac{\pi}{4}=x$ Then ...
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1answer
30 views

How to appraoch solving this series?

I am given the following series and asked to solve it. $\sum\limits_{n=1}^\infty \dfrac{(-2)^n}{(2n+1)!}$ I recognize that this series is somewhat similar to the Taylor series for $sinx$ which is ...
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0answers
38 views

Is there a faster convergence series than the Taylor series?

I am looking for a series expansion which will converge faster than the Taylor series. I mean $$ f(x)=\sum_{n=0}^{N}\frac{f^{(n)}(0)}{n!}x^n $$ For some function you may need large $N$ to get a ...
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2answers
79 views

Verify f'(x) = e^x

The following is a proof I wrote to prove that given $f(x)=e^x$, $f'(x)=e^x$. For this proof we must use the Taylor Series for $e^x$, $\sum\limits_{n=0}^{\infty}\dfrac{x^n}{n!}$. Since the derivative ...
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2answers
40 views

Almost Taylor's Theorem Proof through Integration by Parts

I ALMOST derived Taylor's theorem, which here is $f(x)=\sum_{n=0}^\infty\frac{(x-a)^nf^{(n)}(a)}{n!}$, where $a$ is some arbitrary constant. My attempt: $$f(x)+C=\int f'(x)dx$$ $$\int ...
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0answers
207 views

3Dimensional runge kutta and Euler method ( help to verify the idea and proposition)

Previously,i been discussing this idea with a tutor for sometime. However it turn out that the proof is not comprehensible.But now the derivation for the accuracy up to $O(h^2)$ can be understood. ...
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16 views

Is the following correct way of manipulating taylors series?

For $\sum^{\infty}_{n=1}\frac{(-1)^{n}\pi^{2n}}{4^n(2n+1)!}$. Let $x=\frac{\pi}{2}$, the series becomes ...
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2answers
34 views

How to compute taylor series $f(x)=\frac{1}{1-x}$ about $a=3$?

How to compute taylor series $f(x)=\frac{1}{1-x}$ about $a=3$? It should be associated with the geometric series. Setting $t=x-3,\ x=t+3$, then I don't know how to continue, could someone clarify the ...
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1answer
29 views

How to compute $\sin{(\pi x)}$ about $\frac12$ in taylor series?

The correct answer is supposed to be $\sum\frac{(-1)^n}{(2n)!}\pi^{2n}(x-\frac12)^n$ which I don't understand. Since the function is about $x=\frac12$, so $(x-\frac12)^n$ is good. But ...
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7answers
546 views

How do you create an alternating series with the sign being the same twice in a row?

I am working on a Taylor series question and I have created a series which alternates however, it does so in doubles. in other words it follows the following pattern: $x$, $x$, $-x$, $-x$, $x$, ...
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1answer
85 views

Are there only a few 'universally convergent' Taylor Series?

The Taylor series for $\sin(x)$, centered at any point, converges for all $x$. The Taylor series for $e^{x}$ and $\cos(x)$ do as well. Thus, taking an algebraic function of these (without division) ...
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3answers
37 views

How to compute $\lim\limits_{x\to 0}\frac{e^{x^2}-\cos{(2x)}-3x^2}{x^2\sin{x^2}}$ limits by using taylor series?

How to compute $\lim\limits_{x\to 0}\frac{e^{x^2}-\cos{(2x)}-3x^2}{x^2\sin{x^2}}$ limits by using taylor series? I think that we need to take every familiar taylor series (i.e. $e^x,\sin{x}$) and ...
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1answer
24 views

How to compute the following series using taylor expansion manipulation?

How to compute $\sum^{\infty}_{n=0} \frac{x^n}{(n+2)n!}$ and $\sum^{\infty}_{n=0}(-1)^n \frac{(n+1)x^{2n+1}}{(2n+1)!}$ using taylor expansion manipulation? $1.\sum^{\infty}_{n=0} ...
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2answers
86 views

Characterizations of $e^x$

I've been thinking about the following problem for a while: (AFAIK) the 'exponential function', $e^x$ can be characterized as the unique solution to the following differential equation with initial ...
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1answer
18 views

How to compute taylor series for $f(x)=\frac{1}{1-x}$ about $a=3$ and $f(x)=\sin{x}$ about $a=\frac{\pi}{4}$?

How to compute taylor series for $f(x)=\frac{1}{1-x}$ about $a=3$ and $f(x)=\sin{x}$ about $a=\frac{\pi}{4}$ ? For the first one, using substitution, let $t=x-3$, then $x=t+3$. Then ...
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0answers
14 views

Maclaurin polynomial expansion of $y$ about 1?

Consider the differential equation $\frac{dy}{dx}=2x+\frac{y}{x}$, where $\frac{dy}{dx}=1$ when $x=1$. Find the first three non-zero terms in the Maclaurin polynomial expansion for $y$ about ...
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2answers
37 views

Uncertain how the following step was accomplished.

I'm working through a book example that aims to find the first two nonzero terms of the Laurent expansion of $f(z)=\tan(z)$, about $z=\frac{\pi}{2}$. The substitution $z=\frac{\pi}{2}+u$ is made ...
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3answers
42 views

Use of Taylor series expansion to find second derivative for sixth order method

Use Taylor's expansion to derive sixth order method (i.e $\mathcal{O}(h^6)$) for approximating the second derivative ($f '' (x_0)$ ) for given sufficiently smooth function $f(x)$. I have this things ...
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1answer
25 views

Taylor Polynomial Approximations

I am asked to find a Taylor Polynomial approximation accurate to within $10^{-3}$ for the following function $$f(x)=\frac{1}{x+1}, x \in [-\frac{1}{2},\frac{1}{2}]$$ We know the Taylor expansion for ...
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2answers
25 views

When does Taylor series for g agree with g

For $g(x)=e^{-1/x^2}$ for x not equal to 0 and $g(0)=0$. How to show that the Taylor series for g about 0 agrees with g only at $x=0$? I know that the maclaurin series for g(x) is ...
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2answers
32 views

Check: Radius of Convergence of the Sum of these Complex Taylor Series

I just found the following Taylor series expansions around $z=0$ for the following functions: $\displaystyle \frac{1}{z^{2}-5z+6} = \frac{1}{(z-2)(z-3)} = \frac{-1}{(z-2)} + \frac{1}{(z-3)} = ...
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5answers
108 views

Maclaurin Expansion for $e^{e^{z}}$ at $z=0$

I need to find terms up to degree $5$ of $e^{e^{z}}$ at $z=0$. I tried letting $\omega = e^{z} \approx 1+z+\frac{z^{2}}{2!}+\frac{z^{3}}{3!}+\cdots$, and then substituting these first few terms ...
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Taylor series for $\arctan x$

We use $\frac{1}{1+x^2} = \sum_{n=0}^\infty (-1)^nx^{2n}$, where $|x|<1$ and integration yields $\arctan x = \sum_{n=0}^\infty (-1)^n\frac{x^{2n+1}}{2n+1}$. And by the ratio test this series ...
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1answer
34 views

Find Maclaurin series for integral of $e^{z^2}$

I need to find a Taylor Series expansion of $\displaystyle \int_{0}^{z}e^{\zeta^{2}}d\zeta$ around $z=0$, which shouldn't be hard enough. Except that I can only integrate term-by-term if the Taylor ...
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2answers
78 views

Examine convergence of $\sum_{n=1}^{\infty}(\sqrt[n]{a} - \frac{\sqrt[n]{b}+\sqrt[n]{c}}{2})$

How to examine convergence of $\sum_{n=1}^{\infty}(\sqrt[n]{a} - \frac{\sqrt[n]{b}+\sqrt[n]{c}}{2})$ for $a, b, c> 0$ using Taylor's theorem?
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Prove that the $c_{n}$ in $\frac{1}{1-z-z^{2}}=\displaystyle\sum_{n=0}^{\infty}c_{n}z^{z}$ satisfy a Fibonacci-like recurrence relation

I need to prove that the coefficients $c_{n}$ of the expansion $\frac{1}{1-z-z^{2}}\sum_{n=0}^{\infty}c_{n}z^{n}$ satisfy the recurrence relation $c_{0}=c_{1}=1$, $c_{n}=c_{n-1}+c_{n-2}$, by ...
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2answers
25 views

Taylor series for $\frac{1}{az+b}$ centered at $z=0$ by substitution

I need to find the Taylor series centered at $z=0$ (i.e., the Maclaurin series) for $\displaystyle \frac{1}{az+b}$, where $a,b \in \mathbb{C}$ and $b \neq 0$. I thought it would be good to start out ...
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1answer
41 views

Finding $f^{(12)}(0)$ with $f(x)=\log(e^{x^4}-2x^8)$

Here's how I proceeded: We have $f(x)=x^4+\log\left(1-2x^8e^{-x^4}\right),$ hence for all $x$ such that $-1\le2x^8e^{-x^4}<1$ the following holds: \begin{align} ...
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2answers
52 views

Approximate $\log(1-e^x)$ where $x<0$

The title is pretty self-explanatory, I need to calculate the logit function ($x=\log(p)$): $$x-\log(1-e^x)$$ Where $x<0$, And my problem is to approximate $$\log(1-e^x)$$ I was thinking of ...
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1answer
31 views

Taylor expansion of $\sin(x-y)$

A question asks me to find the partial derivatives of $f: \mathbb{R}^2 \to \mathbb{R}$ with $f(x, y) = \sin(x-y)$ then asks me to give the taylor expansion of $f(\pi/2+h, k)$ in powers of $h$ and $k$ ...
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4answers
66 views

What is the general term for $e^x/(1-x)$

What id the taylor series expansion for $\frac{e^x}{1-x}$? I know that the series expansion for $e^x$ is the sum of $\frac{x^n}{n!}$ from $0$ to $infty$. But how can I account for the $1- x$ in the ...
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0answers
6 views

Expand or approximate entropy of a two-term Gaussian Mixture

Is it possible to create some expansion to approximate this $h(a)$ for $a>0$ near $a\rightarrow0$? $$N(x,v)\equiv\frac{1}{\sqrt{2\pi v}}e^{-\frac{x^{2}}{2v}}$$ $$ ...
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1answer
23 views

How do I work out the validity for a Maclaurin (power) series?

I cannot find the answer to this anywhere so I have decided to make a question. Given a Maclaurin series for a function, how can I quickly work out what the validity is for it? For example, ...
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2answers
69 views

Why Taylor Series or any other approximation method give us approximation of function? Why not give exact equivalent of function?

Lately I am started studying approximation of functions by polynomials and the need for approximation of functions? But what I failed to understand and books did not explain me is that why finding ...
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2answers
53 views

Maclaurin series of $x^3/(e^x-1)$

how would i taylor expand $f(x)=\frac{x^3}{e^x-1}$ around $x=0$? I was thinking of writing $\frac{x^3}{e^x-1}\approx\frac{x^3}{1+x+\frac{x^2}{2}+\frac{x^3}{6}+\frac{x^4}{24}+\dots}$ $~~~~~~~~= ...
12
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1answer
280 views

Convergence of the quadratic map $\left(x-\left(x-\left(x- \dots \right)^2 \right)^2 \right)^2$?

Edit - I changed the title and much of the body to better reflect my full question. The old one I don't really care about, although I appreciate Fabian's answer of course. Here is the plot for ...