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's theorem - C

I've got simple cod for Taylor's Theorem for cosh() function. I'm trying to catch a mistake - the result is about half the real answer. How to do it correctly? ...
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76 views

Show $\ln2 = \sum_\limits{n=1}^\infty\frac1{n2^n}$

Problem: Show that $$\ln2 = \sum_\limits{n=1}^\infty\frac1{n2^n}.$$ My progress: The problem before this one had me find the Taylor series for $\ln(1-x)$ which was $$-\sum\limits_{n=1}^\infty ...
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45 views

Prove that: $\lim \limits_{x \to x_0} \frac{f(x)-T_n(x)}{(x-x_0)^{n}} =0 $

Let $f$ be a function that is differentiable $n$ times at the point $x_0$. Prove that: $$\lim \limits_{x \to x_0} \frac{f(x)-T_n(x)}{(x-x_0)^{n}} =0. $$ It's said that it's Taylor's first ...
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how to evaulate: $\lim \limits_{x \to 0} \frac{\ln(1+x^5)}{(e^{x^3}-1)\sin(x^2)} $

How do I evaluate: $\lim \limits_{x \to 0} \frac{\ln(1+x^5)}{(e^{x^3}-1)\sin(x^2)} $ ? according to Taylor's series, I did like this: $$\lim \limits_{x \to 0} ...
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1answer
80 views

Complex Arctan function and its power series

I face a sequence of confusing questions: In complex plane, note that $arctan(z)$ denote the principal branch of inverse complex tanget function ,by requiring $$\frac{-\pi}{2} < ...
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27 views

what is $\sum_{n=1}^{\infty}\frac{x^{n+1}}{n}-\sum_{n=1}^{\infty}\frac{x^n}{n}$

I posted a question earlier about the taylor of $(1-x)\ln(1-x)$ but i made a miscalculation and decided to delete it, sorry about that. anyways, i solved the miscalculation and i found that ...
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41 views

Calculate with Taylor's series the following limit: $\lim\limits_{x \to \infty}x-x^2\ln\left(1+\frac{1}{x}\right)$

Calculate with Taylor's series the following limit: $$\lim \limits_{x \to \infty}x-x^2\ln\left(1+\frac{1}{x}\right)$$ As I know, I should open an expansion around the point $a=0$, which means using ...
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22 views

Moment List for Standard Normal Distribution

I am stuck trying to find the moment list for a standard normal. I have been told I can find it the similar way for exponential distributions using taylor series. I know the MGF = e^((1/2)(t^2)) for ...
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1answer
40 views

Problem with integral and maclaurin Series

I am working on some problems and I can't seem to figure this one out. The question asks, find/derive the maclaurin series for the following function F(x)= $\int_0^x$$e^{t^2}$dt Here is what I did, ...
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52 views

Is the integral of the sum really the sum of the integrals?

I was asked to find the mclaurin series of $\int_0^x\frac{\arctan (t)}{t}dt$ using the known mclaurin for arctan: $\arctan(t)=\sum_{n=1}^{\infty} \frac{(-1)^{n+1}t^{2n-1}}{2n-1}$ Ok, so what I did ...
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19 views

Generalizate Laurent Series two variable

it is posible make the following series of two variable $$\frac{x^5 y^4 \cos \left(\frac{1}{x}\right)}{x y+1}-x^4 y^3+x^3 y^2+\frac{x^2 y^3}{2}-x^2 y-\frac{x y^2}{2}+x$$ for the serie $$\frac{x \cos ...
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7 views

$f\in C^{\infty}(0,\infty)$ and $f^j(x)=O(x^-a_j)$ with $0\le a_j\lt1$

If $$f\in C^{\infty}(0,\infty)$$ and $$f^j(x)=O(x^-a_j)$$ with $$0\le a_j\lt1$$ then prove that $$f\in C^{\infty}[0,\infty)$$ I do not know how to start to prove this any ideas?
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1answer
31 views

Taylor Expansion for an Inverse fonction

Is there a Straightforward Way to calculate taylor Expansion for the inverse function such as $$ \ e^{\sin x} $$ by knowing the taylor expansion for the function it self, I Think we use the ...
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1answer
51 views

Proposing a numerically sound algorithm for a function as it approaches 0

Suppose I have the function $f(x) = 1 + x - \sin(x)/(x*e^{x})$ I am tasked with proposing a numerically sound algorithm for evaluating f(x) as it approaches 0. My initial thought would be to take ...
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51 views

How to determine the radius of convergence if the Taylor series cannot be written in a neat way?

I am trying to evaluate the radius of convergence of Taylor series centered at zero of function $$f(z)=\frac{\sin(3z)}{\sin(z+\pi/6)}$$ I guess the answer should be $\pi/6$ because the function will ...
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1answer
23 views

Expand $[\frac{\tau^2}{16}(\frac{v}{i+1}+1)^2+\frac{1}{2}]^{-a-2}$

How to expand the expression as showed in the title about $v=0$, in which, $a$ is a natural number and $\tau$ is a real number and $i$ is the complex unit. Many thanks in advance.
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16 views

Calculating Taylor tasks (sinx)

Is there basically anything else behind this task except recognising x is Pi/4 and that it is awfully similar to sinx version of Taylor? First time posting so I probably made some administrative ...
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2answers
19 views

Calculate the improper integral and the Taylor series of $f(x) = \int_0^\infty {e^{-t}\over 1+x \cdot t} \,\mathrm dt$

For the given function: $$f(x) = \int_0^\infty {e^{-t}\over 1+x \cdot t} \,\mathrm dt$$ with $x>0$, calculate the Taylor series of $f(x)$ at $x=0$. I tried different stuff, but I did not get very ...
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2answers
36 views

Solve Limit indeterminate Form using taylor Expansion

It seems that the only way to remove the indeterminate form is to find equivalent of all the functions that pose the problem using taylor Expantion with the right order which is 3 in this case so ...
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1answer
26 views

Asymptotic form when series form of a real analytic function is known

Given an analytic function $f: \mathbb{R} \to \mathbb{R}$ whose Taylor series converges over all $\mathbb{R}$ and is \begin{equation} f(z) = \sum_{k=0}^{\infty}a_k x^k, \end{equation} and where the ...
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34 views

Find the Taylor series expansion of $\csc x$ in ascending powers of $(x-\frac{\pi}{4})$ up to and including the term in $(x-\frac{\pi}{4})^3$

Find the Taylor series expansion of $\csc x$ in ascending powers of $(x-\dfrac{\pi}{4})$ up to and including the term in $(x-\dfrac{\pi}{4})^3$ I've been discussing this with my friend and we're both ...
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57 views

using taylor series to prove $\lim_{{x}\to+{\infty}}e^{-x}=0$ equal to zero without using its algebraic fact.

To be more specific. We know $e^{-x}$=$\sum_{k=0}^{\infty}\frac{(-1)^k}{k!}{x^k}$, without using the fact that $e^{-x}$=$\frac{1}{e^x}$ and using taylor expansion on $e^x$, how do we prove ...
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32 views

Finding the Taylor series about 0 of a function

The function I'm talking about is : $e^\frac{log(1+x)}{x}$ . I can't find a nice closed form for the associated series using just the expansion of $e^x$ . What am I missing here? How can a ...
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172 views

Derivation of the Boltzmann factor in statistical mechanics

I have seen similar derivation of the Boltzmann factor many times before, (http://micro.stanford.edu/~caiwei/me334/Chap8_Canonical_Ensemble_v04.pdf , just for example), which I think is incomplete. ...
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61 views

User defined sine function in python using series expansion

Applying a Maclaurin expansion to the sine function gives ...
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2answers
22 views

Taylor expansion of at a point different from $0$: should the variable be changed?

Find the Taylor expansion of $\arcsin x$ at point $1$. Can we change variable to get the series at point $0$? If yes how, and when do we change again to get back to $1$? More generally Let's ...
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81 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 ...
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209 views

Summing up series which is similar to Taylor expansion

I am given series as $$S= \frac{1.2.3}{1!} + \frac{2.3.4}{2!} + \frac{3.4.5}{3!}+.........$$ I know this series looks similar to $e^x$ expansion and probably $x=1$ here so how to express my series ...
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1answer
49 views

Taylor's theorem on manifold

Taylor's theorem for real-valued functions on manifolds is straightforward, and doesn't even require anything beyond differential structure. How does Taylor's theorem work for manifold-valued ...
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108 views

Unknown Taylor expansion

I have come across a few apparently related Taylor expansions, as detailed below: \begin{align} &\dots\frac{a^7}{140}-\frac{a^6}{80}-\frac{3 ...
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1answer
23 views

Matrix Taylor series

I'm just wondering why $$(A+\epsilon)^{-1}=A^{-1}-A^{-1}\epsilon A^{-1}+\mathcal O(\epsilon^2).$$ Can someone please show me the steps? Like in the second term, why is $\epsilon$ sandwiched between ...
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1answer
31 views

Exponential of little o

I have a series and the error is of order $o(x^N)$: $f(x)=\sum_1^N x^n+ o(x^N)$ Now I want to take exponential on both sides. What happens with the error part?
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29 views

Taylor expansion of $\ln(1+ae^{bx})$?

As we know, $$\ln(1+x)=\sum_{n=1}^{\infty}\frac{(-1)^{n+1}x^n}{n}$$ when $|x|<1$, but what for a function $\log(1+ae^{bx})$? can we use it here?If not, then how'll we expand it?
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2answers
36 views

What is the point of the remainder term in the Taylor series

Take the first order taylor approximation: We say $f$ is differentiable at $x_0 \in S^o$ if $\exists \space \nabla f(x_0)$ and a function $R: \mathbb{R}^n \to \mathbb{R}$ such that $f(x) = f(x_0) + ...
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5answers
55 views

Taylor expansion of $\log(3+x)$

Using the standard result of $\log$ find the taylor expansion of $$\log{(3+x)}$$ Now I believe $$\log{(1+x)} = \log{(1+x)} = \sum^{\infty}_{n=1} \frac{(-1)^{n+1}}{n}x^{n}$$ So to find $\log{(3+x)}$ ...
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1answer
16 views

Convergence radius argument

I'm studying complex function theory and I ran into this argument made by my prof but I can't really wrap my head around it. Set $f(z):=\frac{1}{7+z^2}$ Now notice $\sum_{n=0}^\infty ...
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1answer
60 views

Taylor series at two different points

If I have a function $y(x)$ for which there is a Taylor series about $x=1$ that has an infinite radius of convergence, and I also have a Taylor series for $y(x)$ about $x=0$ (with unknown radius of ...
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18 views

Function property of $o$

I have a doubt regarding $o$-function. Could we write $o(\|\theta h)\|)=\theta \ o(\|h\|)$ ?
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35 views

calculate limit with Taylor and L'hopital

i need to calculate a limit, and our teacher told us to use L'hopital and Taylor aproximations. $$\lim_{x\to 0}\left({\sin(x)\over x}\right)^{1/x^2}$$ and that must be equal to; $$e^{1/6}$$ and i ...
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Estimating $\int_0^1\sin(x^2)dx$ using the Taylor expansion of $\sin(x)$

Problem: a) Find the Taylor polynomial $T_6(x)$ for $f(x) = \sin(x)$ about $x=0$. I found this to be $x-\frac{x^3}{6} + \frac{x^5}{120} + O(x^6)$. b) Use this to estimate $\int_0^1\sin(x^2)dx$ with ...
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1answer
148 views

taylor expansion in cylindrical coordinates

If I have a function Y($r$,$\theta$) in cylindrical polar coordinate system, then how do I Taylor expand this function around some point ($r_0$,$\theta_0$)? I want the exact formula for Taylor ...
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60 views

Taylor series applied to logarithm

Following formula at $k=0$ yields $\frac{1}{2i}\log\frac{1+i}{1-i}=\frac{\pi}{4}$. $$\sum_{n=0}^\infty\frac{i^{2n}}{2(n+2k)+1}$$ At $k\in\Bbb N$, Mathematica throws out $\frac{1}{2} ...
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24 views

Binomial random walk: Taylor series

Consider the following probabilities associated with a binomial random walk: $$ p(y',t') = \dfrac{1}{2}p(y' + \delta y, t' - \delta t) + \dfrac{1}{2}p(y' - \delta y, t' - \delta t) $$ What's ...
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58 views

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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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 ...