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 $(x+2)/(2-3x)$ at $x=2$ [closed]

How can I find Taylor series for $$\frac{(x+2)}{(2-3x)}$$ at $x=2$?
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1answer
30 views

Taylor series $\ln(x+3)$ at $x=1$

Taylor series $\ln(x+3)$ at $x=1$ I am a little confused if both ways are correct: $y=x-1$ $$\ln(y+4)=\ln(4) + \ln (1+y/4)=...=\ln(4)+\sum_{n=1}^\infty(-1)^{n-1}(1/4)^n\frac{(x-1)^n}{n}$$ or ...
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1answer
28 views

An Example of Taylor Series

Consider a positive function $f(x)$ and suppose that we would like to approximate its value around some point $x_0$. One way to do so is to use two-term Taylor series expansion as follows. $$ f(x) ...
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1answer
24 views

Taylor expansion of $\frac{(-1)^n}{\ln n(1+\frac{1}{n\ln n}+o(\frac{1}{n\ln n})}$

I can't get the right terms: $$\frac{(-1)^n}{\ln n + \frac{(-1)^n}n + o(\frac1n)}=\frac{(-1)^n}{\ln n} - \frac1{n\ln^2 n}+o\left(\frac1{n\ln^2 n}\right)$$ My thoughts $$\frac{(-1)^n}{\ln ...
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1answer
33 views

How can get the series of $\log(x/(x-1))$ at $x=\infty$

When I used the Wolfram to give me the Taylor series of $\log(x/(x-1))$, I was amazed of the result. The Wolfram give me a Laurent series at $x=\infty$ as follow ...
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1answer
38 views

Where does the $\mathcal{O}$-term come from in Taylor series?

I understand Taylor series in general, but I've always been a bit uncertain about the $\mathcal{O}$-term, when I see it used in Taylor series. For example in one of my study materials I have the ...
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1answer
43 views

What is the remainder of $|e-\sum_{j=0}^n{1\over j!}|$?

I have to find the smallest $n$ such that $|e-\sum_{j=0}^n{1\over j!}|<0.001$, but I want to do it with the remainder. I know that it is ${e^c\over (n+1)!}$ where $0<c<1$, but how do you get ...
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1answer
41 views

Find the smallest $n$ such that $|e-\sum_{j=0}^n{1\over j!}|<0.001$.

Find the smallest $n$ such that $|e-\sum_{j=0}^n{1\over j!}|<0.001$. I know it has something to do with remainder, or Taylor expansion, but I am week in this material. What I did is merely: ...
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0answers
60 views

Evaluating $\lim_{n\to\infty}n\sin(2\pi en!)$ [duplicate]

So I need to evaluate $$\lim_{n\to\infty}n\sin(2\pi en!)$$ And yes, I know it was discussed here and similar limit was discussed here, but I didn't feel quite okay with the solutions/hints given in ...
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2answers
47 views

Find $\lim_\limits{n\to\infty}\sin(\pi\sqrt[3]{n^3+1})$.

Find $\lim_\limits{n\to\infty}\sin(\pi\sqrt[3]{n^3+1})$. I am trying to find it using Taylor series. What I did so far is: $\sqrt[3]{z+1}=1+O(z)$ (I really can't tell when I should be done ...
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3answers
33 views

Find taylor of $\psi (z)$ where $(e^z-1)^2=z^2 \psi (z)$ - first 3 terms

I was asked to find the first three terms in the taylor series of $\psi (z)$ around $z=0$ where $(e^z-1)^2=z^2 \psi(z)$ and I'm having a few difficulties. My original idea was to say $\psi ...
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1answer
156 views

Evaluate $\pi$ using $\arctan(\frac{\sqrt{3}}{3})$

I Have to evaluate $\pi$ using $\arctan(\frac{\sqrt{3}}{3})$ with an error with no more than $10^{-10}$ using taylor approximation $ p_{2n-1}(x) \approx\arctan(x)$ . So, After manipulation, I get: ...
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0answers
40 views

Proving $f=0$ if $f({1\over k})=0$ $\forall k\in \Bbb{N}$ .

Let $f\in C^{\infty}[-1,1]$ and let $M$ be a constant such that $|f^{(j)}(x)|\le M$ $\forall j\in \Bbb{Z}_{+}$ and $x\in [-1,1]$. Prove that if $f({1\over k})=0$ $\forall k\in \Bbb{N}$ then $f=0$. I ...
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2answers
34 views

MacLaurin Series with Variable in Denominator

A friend of mine was talking about how finding MacLaurin series for functions with variables in the denominator might prove difficult without tables. We started making lots of crazy problems, but one ...
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2answers
31 views

Poisson complete statistic

I have the same question as this thread, but I cannot understand the proof. The problem is, given $f(\lambda)=\sum_{k=0}^\infty g(k)\frac{(n\lambda)^k}{k!}=0,\forall\lambda>0$. How to show ...
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4answers
72 views

Find $\lim_\limits{x\to 1^-}{\ln x\cdot \ln(1-x)}$.

Find $\lim_\limits{x\to 1^-}{\ln x\cdot \ln(1-x)}$. I can't even start because I don't really know what $x\to 1^-$ means. If you know what it means it would really help me. I would as well if you ...
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3answers
302 views

How to calculate $\exp(-x)$ using Taylor series

We know that the Taylor series expansion of $e^x$ is \begin{equation} e^x = \sum\limits_{i=1}^{\infty}\frac{x^{i-1}}{(i-1)!}. \end{equation} If I have to use this formula to evaluate $e^{-20}$, how ...
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0answers
19 views

Coefficient in Taylor Series expansion [duplicate]

Find the coefficient of $(z-\pi)^2$ in the Taylor series expansion around $\pi$ if $$f(z) = \begin{cases} \frac{\sin z}{z-\pi} & \quad, z \neq \pi \\ -1 & \quad, z=\pi \end{cases}$$
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1answer
27 views

Number of derivatives in a taylor series expansion

I would like to confirm if the number of derivatives we need to calculate in a specific order of a taylor series expansion is the sum of the multinomial coefficient of that order: $$ f:\mathbb{R}^k ...
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1answer
62 views

Evaluating $\lim_{n \to \infty} \sum\limits_{k=1}^n \frac{1}{k 2^k} $

Question: How to compute $$ \lim_{n \to \infty} \sum\limits_{k=1}^n \frac{1}{k 2^k}? $$ Here is what I have tried so far: Define $s_n=\sum\limits_{k=1}^n \frac{1}{k 2^k}$ for every index $n$, ...
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1answer
32 views

Taylor theorem for f(x+h)

I am following a proof that applies Taylor's theorem on this document (http://www.gautampendse.com/software/lasso/webpage/pendseLassoShooting.pdf) I am not understanding one of the terms explained on ...
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16 views

A simple question about Delta Method's demonstration.

Suppose that $\sqrt{n}(X_n-\mu)\stackrel{D}{\longrightarrow}X$ and consider $g:\mathbb{R}\rightarrow\mathbb{R}$ a function such that first derivative $g'$ is continuous in a neighbourhood of $\mu$, ...
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3answers
53 views

taylor of $\frac{1}{z}$ at $a=-2$

I want to find the taylor series representation of $f(z)=\frac{1}{z}$ at $a=-2$. The point of this exercise is not to find some pattern in the derivatives, infact we are not meant to find any ...
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4answers
106 views

Find $\lim_{n\to \infty}(\cos{x\over n})^{n^2}$

Find $$\lim_{n\to \infty}\left(\cos{x\over n}\right)^{n^2}$$ where $x\in \Bbb{R}.$ I tried using taylor series. A complete mess, and an area I am not very good at. I tried using $e$ which also gave me ...
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1answer
31 views

nth derivative of a troublesome function

I don't know where to start on this problem. I'm trying to get the 2015th derivative(at x = 0) of f(x) = x^2 * arctan(x). Doing the derivatives one by one seems a little troublesome... What do you ...
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2answers
57 views

What is the coefficient of $(z-\pi)^2$ in Taylor series expansion of $\sin (z)/ (z-\pi)$

I want to determine the coefficient of $(z- \pi)^2$ in Taylor series expansion of $f(z)=\sin (z)/ (z-\pi)$ if $z \neq \pi $, $-1$ if $z=\pi$ around $\pi$. How can this be done? I don't know how to do ...
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1answer
53 views

Sum of exponential square series

I have a infinite sum which I wonder if it will converge to a simpler function $f(r) = \Sigma_n r^{n^2} , r<1$, I also interested in case $r$ is a complex number on unity circle $r = ...
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2answers
80 views

Higher Order Terms in Stirling's Approximation

Some websites and books give stirling approximation as $$n! = \sqrt{2 \pi n} \left( \frac{n}{e} \right)^n \left( 1 + O \left(\frac{1}{n} \right)\right)$$ However when I check their derivations most ...
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1answer
41 views

Find the coefficient of $x^{4}$ from $(1+x)^{1/3}$

Find the coefficient of $x^{4}$ from $(1+x)^{1/3}$ Should I use the formula $C(n,k) = n!/[k!(n-k)!]$? And what is the solution of this problem?
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1answer
24 views

taylor series approximation of e function

in the equation $$e^{y(x)}=1+2x-\frac{y(x)}{1-x}$$ $y(0)=0$ because using the taylor series and by comparing the coefficients we obtain $$1+y(0)=1-y(0)$$But why is using the taylor series allowed. ...
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1answer
41 views

Functionals' Taylors Theorem

Consider functional $F:B\to \mathbb{R}$, where B is a Banach space eg. $B=H^{1}(\mathbb{R}^{d},\mathbb{C})$. Then Taylor's theorem for functionals is: Suppose that the line segment between u ∈ ...
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1answer
29 views

Maclaurin Expansion of $\ln(1+4x^2+4x)$ in terms of $\sum a_k x^k$

Maclaurin Expansion of $\ln(1+4x^2+4x)$ in terms of $\sum a_k x^k$ The question has written to $x^2$ term before the $x$ - does that have anything to do with how to solve the problem? Am I meant to ...
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1answer
54 views

Taylor expansion in $p$-adic integers

Let $f \in Z_p[X]$, then for $ x, y \in Z_p$, $\exists a \in Z_p$ s.t. $f(y)=f(x)+(y-x)f'(x)+(y-x)^2a$. Why is Taylor formula applicable to polynomial in $p$-adic integers $Z_p$? What condition ...
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1answer
43 views

Extending Taylor's theorem from one to several variables

In my calculus class we are dealing with Taylor´s theorem in several variables. When we were looking at the function $f(x,y)=\sin(xy)$ my teacher said that instead of applying the theorem in several ...
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1answer
54 views

Using $\ln (\cos x)=\frac{-x^2}{2}-\frac{x^4}{12}+…$, approximate $\ln 2$ in terms of $\pi$

Using $f(x)=\ln (\cos x)=\dfrac{-x^2}{2}-\dfrac{x^4}{12}+\dots $, approximate $\ln 2$ in terms of $\pi$. I know $\cos(x)$ will never be two - so what can I actually substitute in to get something ...
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27 views

Taylor polynomial to find an approximation

Use the Taylor polynomial of degree 5 to give an approximation for ln(2) This may seem really simple but I have no idea how to do it, please help.
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1answer
30 views

Is the taylor polynomial of degree $2$ near $(0,0)$ of $𝑓(𝑥, 𝑦) = \frac{1}{ 2 - (𝑥 + 𝑦^2)}$ the following:

$ P(𝑥, 𝑦) = \frac{1}{2} + \frac{𝑥}{4} + \frac{𝑥^2}{4} + \frac{𝑦^2}{2}$ Is this right? I can't tell, as I can't seem to see the remainder going to $0$ when divided by $x^2 + y^2$ as $(x, y) → ...
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1answer
136 views

Can $f(g(x))$ be a polynomial?

Let $f(x)$ and $g(x)$ be nonpolynomial real-entire functions. Is it possible that $f(g(x))$ is equal to a polynomial ? edit Some comments : I was thinking about iterations. So for instance ...
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1answer
26 views

Functional expansion

I am confused by this expansion in Landau and Lifshitz: First, they define $\textbf{v}' = \textbf{v} + \textbf{$\epsilon$}$. So for a function $L$, $$L(v'^2) = L(v^2 + ...
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2answers
34 views

Another question about $x_0$ in the Taylor series

When we talk about Taylor series, we say it's around point $x_0$. It's in the Taylor series formula: $$f(x) = f(x_0) + (x-x_0)f'(x_0) + \frac{(x-x_0)^2f''(x_0)}{2} + \frac{(x-x_0)^3f'''(x_0)}{6} + + ...
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0answers
16 views

How can I show the remainder of this Taylor polynomial $R(h)/h^2$ goes to $0$ as $h$ goes to $0$?

Given the function $f(x, y) = 1/(2 - x - y^2)$ I found that the second-degree Taylor polynomial is $P(x, y) = 1/2 + x/4 + x^2/4 + y^2/2$ How can I show the remainder $R(x, y) = f(x, y) - P(x, y)$ ...
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1answer
55 views

Taylor series expansion of the function $f(x)=x \arctan x-0.5 \log(1+x^2)$ about the origin int the region {$|x|\le1$}

Find the Taylor series expansion of the function $\color {green}{f(x)=x \tan^{-1} x-0.5 \log(1+x^2)}$ about the origin int the region {$|x|\le1$} My effort: I know $\displaystyle \log ...
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1answer
39 views

Physically, what meaning have Taylor series which have their lower order terms equal to zero, but their higher order terms non zero?

Usually, when using a Taylor series to describe a function (which may itself be a model of some physical phenomenon), we often throw out the higher order terms, as they are quite small relative to the ...
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2answers
34 views

On finding the order of an infinitely small quantity

Given an infinitely small quantity: $$\alpha \left ( x \right )= \tan \left ( x \right )-\sin \left( x \right)$$ as x aproaches $0$, and computing the corresponding asymptotic relationship. What does ...
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0answers
29 views

Name of method which includes Taylor linearization inside fixed point iteration

I read paper about Horn-Schunck multiscale method for computing optical flow Core part of this algorithm is minimizing some functional. One part of functional contains nonlinear term inside L2 norm. ...
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1answer
51 views

Taylor expansion of $1/(1+z)$

How do I obtain the Taylor expansion of $$\frac{1}{1+z}$$ about $a=i$ please? Do I just expand the series using the binomial expansion?
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3answers
40 views

Taylor series of $\ln(1+x)$

So let's say we want to obtain the Taylor series for $\ln(1+x)$. We know that its derivative is $\dfrac{1}{1+x}$, which has the series $\sum_{n=0}^{\infty} (-1)^nx^n$. Can we take the antiderivative ...
3
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1answer
26 views

using Taylor's Theorem to find region of convergence of series

!(http://imgur.com/0fDL4KZ) I am a third year Electrical engineering student, and I was going through one of the example from my math module lecture notes but couldn't understand the solution printed ...
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1answer
23 views

Find a polynomial $Q$ of degree $k$ and a remainder function $E$ for $f(x)=\frac{1}{1-x}$.

There is a theorem in our textbook saying that rather than calculating all the derivatives needed to compute the taylor polynomial, if one can find, by any means, a polynomial $Q$ of degree $k$ such ...
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0answers
27 views

complex taylor series

I have a series $ f(z)= 1 - z + z^2 - z^3 $ and i want to substitute $ z=b + (z-b) $ into the equation, (where $b=1/2+i/2$) and find the first two coefficients. Wont they just be the same as before ...