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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5
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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 $u=\arctan{(x)}$ ...
1
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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: ...
0
votes
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 ...
1
vote
0answers
37 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 ...
0
votes
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 ...
0
votes
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 ...
2
votes
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 ...
0
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0answers
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 ...
2
votes
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 ...
1
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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 ...
2
votes
2answers
89 views

Series expansion for integral including error function

$\DeclareMathOperator{\erfc}{erfc} \DeclareMathOperator{\Ei}{Ei} $ What is the series expansion of $f$ for small $q$? \begin{align} U(q) &= q e^{q^2}\erfc q\\ I(q,q') &= ...
0
votes
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 ...
0
votes
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 ...
7
votes
2answers
97 views

How can I get f(x) from its Maclaurin series?

I know how to get a Maclaurin series when $f(x)$ is given. I have to find $\sum_{n=0}^{\infty}\frac{f^{(k)}(0)}{k!}x^k$. But how can I get $f(x)$ from its Taylor series? The problem is $$f(x) = ...
2
votes
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} ...
0
votes
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 ...
2
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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 ...
4
votes
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: $x$, $x$, $-x$, $-x$, $x$, ...
1
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2answers
36 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 ...
0
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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 ...
0
votes
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 ...
0
votes
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 ...
5
votes
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 ...
5
votes
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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0answers
40 views

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 ...
1
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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)} = ...
1
vote
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 ...
5
votes
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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3answers
33 views

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 ...
0
votes
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 ...
0
votes
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} ...
0
votes
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 ...
0
votes
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 $k$ ...
1
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1answer
36 views

Find the first two terms in the perturbation expansion of the solution

I want to find the $\mathcal{O}(1)$ and $\mathcal{O}(\epsilon)$ terms in the pedestrian expansion $y = y_0 + \epsilon y_1 + \epsilon ^2 y_2 + \dots$, where $y$ satisfies the following second order ...
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votes
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 $a\rightarrow0$? $$N(x,v)\equiv\frac{1}{\sqrt{2\pi v}}e^{-\frac{x^{2}}{2v}}$$ $$ ...
1
vote
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 ...
0
votes
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, ...
0
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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 $\frac{x^3}{e^x-1}\approx\frac{x^3}{1+x+\frac{x^2}{2}+\frac{x^3}{6}+\frac{x^4}{24}+\dots}$ $~~~~~~~~= ...
0
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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
48 views

Why does the taylor series of $\frac {1}{\ln x}$ have a non-infinite radius of convergence?

Shouldn't the taylor series of a function be equal to that function for any input value? Why does this not work for the taylor series of $\frac {1}{\ln x}$ when $|x| \gt 1$? Edit: I do mean the ...
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3answers
34 views

Taylor's series and ln

Can someone explain to me how to find the $\lim \limits_{x \to 3} \frac{\ln|4-x|}{x-3}$ using taylor's series. Can someone explain the proof of $\ln|4-x|$ to power series please
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0answers
145 views

Is there a way to write this recurrence relation in a simpler way (or a way that's faster to program)?

I have the following recurrence relation for some coefficients $$b_{n+2} = \frac{1}{(n+3)(n+2)P_0} \sum_{k=1}^n (n-k+2) (n-k+1) b_k b_{n-k+2}, \quad n>1$$ with $b_1$ to $b_3$ and $P_0$ being the ...
0
votes
0answers
22 views

a function with many branch points : the radius of convergence of its Taylor series

How can I be convinced that if a (locally holomorphic) function $f(z)$ has many branch points, say at $\beta_1,\beta_2,\ldots,\beta_n,\ldots$, and all of the weirdest type, then the radius of ...
0
votes
1answer
21 views

Radius of Convergence of Taylor series without finding the series

How do you find the radius of convergence of a Taylor series for a function centered at point $z_0$ without actually finding the Taylor series? I know that we can use comparison test, ratio test or ...
0
votes
1answer
25 views

Expanding $1/z$ about $z=-1$ using Taylor series vs Power Series

I need to expand $1/z$ about $z_0=-1$. I decided to do it using both methods, which don't agree. Using Taylor: Finding coefficients: $$f^{(n)}(z)=(-1)^n n!/z^{n+1} \Rightarrow f^{(n)}(-1)=-n!$$ ...
3
votes
2answers
34 views

Exponential Taylor series with $k$ step

It is well-known that $$\sum_{n=0}^\infty \frac{x^n}{n!} = e ^x$$ or $$\sum_{n=0}^\infty \frac{x^{2n}}{(2n)!} = \cosh x $$ My question is what we know about the sum for arbitrary $k \in \mathbb{N}$: ...
1
vote
1answer
34 views

Complex Taylor Series by substitution

I need to find the first few terms or so of the Taylor series centered at $z_0 = 0$ regarding these functions: a) $e^{z\sin z}$ b)$(1+z)^z = e^{z \ln (1+z)}$ c)$\cos (1 + z^3) $ d) $e^{e^z}$ ...
0
votes
0answers
15 views

Value of Taylor Approximation

I'm studying Trust Region Methods and there's one simple thing I'm finding it hard to wrap my head around. The premise of the method is to approximate $f(x)$ as a quadratic model: ...
2
votes
3answers
41 views

$n$-th Term for Maclaurin Series

On a Calculus BC test I had this morning, I had to find the first five terms and the $n$-th term of the following function: $$ f(x) = x \cos(3x)$$ According to my instructor, I could've manipulated ...
2
votes
1answer
33 views

How do I expand this function around zero?

The function is $$ \sqrt{\frac{\sin(x)}{x}} $$ I need to expand it to the order $x^2$ around $0$. The solution is supposed to be: $$ 1-\frac{x^2}{12}+\mathcal{O}(x^4) $$ How do I proceed?