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.
1
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1answer
25 views
Finding power series for $f(x) = \frac{4x+53}{x^2-x-30}$
Given $f(x) = \dfrac{4x+53}{x^2-x-30}$, display it as a power series and find the radius of convergence. then calculate $f^{(20)}(0)$
So what I did was look at the Taylor Series Formula:
$$f(x) = ...
0
votes
2answers
61 views
Taylor series of $f(x)=\frac {e^x-1}{x}$
I am asked to expand $f(x)=\frac {e^x-1}{x}$ centered at 0 using the known Talyor series of functions.
How to simplify the function so that it can be expanded more easily?
1
vote
1answer
40 views
Is $e^z\sum_{k=0}^\infty\frac{k^3}{3^k}z^k$ analytic inside $|z|=3$?
Am I correct that the following function is analytic at least inside $|z|=3$? (I used the ratio test.) The solutions manual says that the function is analytic on and inside |z|=1, so I wonder if I'm ...
0
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0answers
40 views
funcitonal series convergence… SOS… [duplicate]
Is it possible to show that $f(x) = \sum_{k=0}^\infty (-1)^k \frac{x^k \sqrt{k}}{k!}$ converges even when $x\rightarrow \infty$ ?
i.e. As a function of x, does $\lim_{x\to\infty} f(x)$ exist or at ...
1
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0answers
26 views
optimal control -Taylor expansion - PDE problem
I am trying to follow perturbation analysis in this paper (Optimal control of fluid limits of queuing networks and stochasticity corrections) and I am stuck at one point.
For the given control ...
-2
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0answers
40 views
Solve $ z= \frac{2a^2}{-4+z} + \frac{2a^4}{(-4+z)^2(-16+z)} + \cdots $.
I am trying to solve $z\in \mathbb{C}$ in terms of $a\in \mathbb{C}$, where
$$
z= \frac{2a^2}{-4+z} + \frac{2a^4}{(-4+z)^2(-16+z)} + \cdots.
$$
I plugged $z= \sum_{k=0}^\infty c_k a^k $ into the ...
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0answers
42 views
Maclaurin series expansion of an expression that involves a fraction
In the context of statistical mechanics the "classical trace" is defined as $Tr(A e^{-\beta H}) = \int dr^N dp^N A e^{-\beta H}$ where $r^N$ and $p^N$ are phase space variables. So if $\Delta H$ is a ...
1
vote
1answer
145 views
An issue with approximations of a recurrence sequence
By trying to give an approximation to a given recurrence sequence I encountered a problem.
To be more precise I have a method but it fails if the right condition is not met and I wonder how I should ...
0
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1answer
269 views
taylor expansion of exponential function
To prove CLT of binomial distribution,
$$X \sim \mbox{bin}(n,p)$$
$M_X(t)=(p e^t+q)^n$ where $M$ is mgf.
Let $Z=\frac{X-np}{ \sqrt{npq}}$, $\sigma =\sqrt{npq}$, then
$$
\begin{align}
...
2
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3answers
37 views
Finding the Taylor series of $f(z)=\frac{1}{z}$ around $z=z_{0}$
I was asked the following (homework) question:
For each (different but constant) $z_{0}\in G:=\{z\in\mathbb{C}:\,
z\neq0$} find a power series $\sum_{n=0}^{\infty}a_{n}(z-z_{0})^{n}$
whose sum ...
0
votes
2answers
29 views
Series expansion with remaining $log n$
I'm studying the asymptotic behavior $(n \rightarrow \infty)$ of the following formula, where $k$ is a given constant.
$$ \frac{1}{n^{k(k+1)/(2n)}(2kn−k(1+k) \ln n)^2}$$
I'm trying to do a series ...
0
votes
1answer
27 views
Taylor series with function composition
Pretty simple, but I want to take the first order taylor series expansion of the following:
$f(g(x,y+Δy))$
Would the following be correct?
$f(g(x,y+Δy)) = f(g(x,y) + \frac{\partial}{\partial ...
1
vote
3answers
59 views
Taylor Polynomial for $x^{1/3}$
a. Compute the Taylor polynomial $T_3(x)$ for the function $(x)^{1/3}$ around the point $x=1$.
b. Compute an error bound for the above approximation at $x = 1.3$.
I'm having trouble figuring ...
5
votes
4answers
133 views
Are Taylor series and power series the same “thing”?
I was just wondering in the lingo of Mathematics, are these two "ideas" the same? I know we have Taylor series, and their specialisation the Maclaurin series, but are power series a more general ...
1
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0answers
47 views
Taylor Expansion of Power Series
Suppose that $\space f:[0,1]\rightarrow \mathbb{R}$ is real analytic and that its power series expansion is:
$\\ f(x)=\sum\limits_{n=0}^\infty a_nx^n$
Prove that there exists an $x_0\epsilon (0,x)$ ...
2
votes
2answers
1k views
Multiplying Taylor series and composition
I have two questions:
A. I know the taylor series for $\arctan(x)$ and for $e^x$. How do I find the series for $\arctan(x)\cdot e^x$ ?
B. Say I want to find the series for $\arctan(g(x))$, do I just ...
5
votes
0answers
65 views
Maclaurin series of $f(x)=\sinh(1/x)$?
As we know the formula of Maclaurin series for $f(x) = \sinh(x)$ is $f(x)=x+x^3/3! + x^5/5!+\ldots$
Could anyone tell me what is the Maclaurin series of $f(x)=\sinh(1/x)$?
2
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0answers
33 views
Taylor series of a vector field?
Can someone give me the definition of the Taylor Series of a vector field in $\mathbb{C}^2$?
Thanks!
1
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1answer
56 views
Inverse Function Thorem
Let $f,g:\mathbb R\to\mathbb R$ be smooth functions with $f(0)=0$ and f'$(0)\neq 0$. Consider the equation $f(x)=tg(x), t\in \mathbb R$.
Show that in a suitable small interval $|t|\leq \delta$, there ...
0
votes
1answer
70 views
Finding the error of the Taylor expansion of $\log(1 + x)$
The questions is as defined below.
Let $f(x)= \log(1+x)$. Show that the Taylor remainder $R_{0,k}(x)$, defined by $$R_{a,k}(x)= f(a+x) - P_{a,k}(x) = f(a+h) -\sum_{j=0}^{k} \frac ...
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0answers
25 views
Binomial Expansion problem error
I tried solving this question but failed.
a) Expand $(1+2x)^{1/4}$ in ascending powers of $x$ up to and including the term in $x^3$, simplifying each term as far as possible.
b) By substituting ...
3
votes
1answer
57 views
Determine the Maclaurin series expansion of $\frac{\mathrm{exp}(z)}{(z+1)}$
Determine the Maclaurin series expansion of $\frac{\mathrm{exp}(z)}{(z+1)}$.
This is the composition of the series expansion of the exponential function centered about $z = -1$. We can rectify the ...
0
votes
0answers
18 views
Estimate the degree of a Taylor Polynomial using its Error Term
In my 2nd year studying Maths at Uni and revising for a Numerical Analysis final exam. We're given 1 past paper but no solutions, and I can't answer this question:
Use the error term of a Taylor ...
1
vote
1answer
166 views
Complex Analysis Taylor Series
So the problem states:
"Say f(z) := log z is the principal branch of the logarithm (the primitive of 1/z on the region C(-infinity,0]). Show that the Taylor series of f(z) about $z_0 = -1 + i$ takes ...
15
votes
1answer
205 views
$x^3-3x-3=0$, prove that $10^x<127$
$x$ is the real root of the equation $$3x^3-5x+8=0,\tag 1$$ prove that $$e^x>\frac{40}{237}.$$
I find this inequality in a very accidental way,I think it's very difficult,because the actual value ...
5
votes
1answer
62 views
Taylor series and tempered distributions
Suppose we have a function $\psi$ in $\mathbb{R}$ other than a polynomial which is equal to its Taylor Series for every point in $\mathbb{R}$. Why is the following statement valid?
When we interpret ...
1
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1answer
80 views
Laurent expansion of $\csc^2(\frac{\pi}{z})$ about $\frac{1}{3}$ for $|z-\frac{1}{3}| \lt \frac{1}{12}$
The question is: Find the Laurent expansion of $\csc^2(\frac{\pi}{z})$ about $\frac{1}{3}$ for $|z-\frac{1}{3}| \lt \frac{1}{12}$. In particular what is the coefficient of $(z-\frac{1}{3})^{-2}$.
I ...
2
votes
2answers
58 views
How does one get the Bernoulli numbers via the generating function?
Here is the definition:
Bernoulli numbers arise in Taylor series in the expansion $$\frac{x}{e^x-1}=\sum_{k=0}^\infty B_k \frac{x^k}{k!}$$
I've tried to naively expand $\frac{x}{e^x-1}$ around ...
0
votes
1answer
51 views
Itō's Lemma neglecting terms
In my project I am trying to give a Heuristic proof of Itō's lemma. I show $E[dW_t^2] = dt$
I take $g(x,t)$ to be a twice continuously differentiable function and $dt$ to be infinitesimally small.
...
1
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1answer
48 views
Integral remainder converges to 0
I want to show that
$\displaystyle \log(1+x)=\sum_{n=1}^{\infty}(-1)^{n-1}\frac{x^n}{n}$
for $-1<x\leq1$
i want to show it with the integral remainder of the taylor series that gave me:
...
0
votes
2answers
66 views
taylor expansion of an integral $\int_0^1{e^{x^2}}$
I need to calculate $\int_0^1{e^{x^2}\:dx}$ with taylor expasin in accurancy of less than 0.001. The taylor expansion around $x_0=0$ is $e^{x^2}=1+x^2+\frac{x^4}{3!}+...$. I need to calculate when the ...
1
vote
1answer
48 views
Prove that d/dx (sin x) = cos x, using Taylor series
Show by differentiation of the series for sin x that
$$\frac{d}{dx} (\sin x) = \cos x$$ (Using Taylor series.)
If you can given an indication or solved answer to my question would be great.
Thanks ...
2
votes
0answers
39 views
Determine the series for cos x^2
Use the series for Cos x (Taylor Series)
If you could give me help or the solution to the problem, that would be great!
Thanks
1
vote
1answer
33 views
Points around which one expands and the radiuses of convergence
I'm trying to make sense of the following passage:
Let $f(x)=\frac{1}{x+1}$ and $R_0$ the radius of convergence of the Taylor series of $f$ around $x_0=0$, analogously: $R_1$ — around ...
4
votes
3answers
55 views
Taylor Series for $e^x$ where $x = 1$, estimating the Error
I'm trying to calculate $e$ to a certain number of digits. The Maclaurin Series expansion of $\displaystyle e^x = \sum_{n=0}^\infty \frac{x^n}{n!}$. When $x = 1$ we can approximate the value of $e$ by ...
1
vote
2answers
41 views
Multiplication of two Taylor expansions
I'm trying to calculate a Taylor expansion which is : $\cos(x). exp(x)$ in the neighborhood of 0 in order 3
this is the result I got :
$$\cos(x). exp(x) = ...
6
votes
3answers
60 views
In Taylor series, what's the significance of choosing the point of expansion $x=a$?
So I read about the Taylor series and it said you can choose to expand the series around a given point ($x=a$). Does it matter which point you choose in calculating the value of the series?
For ...
2
votes
2answers
63 views
A question on the convergence of a Taylor series of some prominent function
The function $f:\mathbb{R}\to\mathbb{R}$ defined by
$$
f(x)=\begin{cases}e^{-\frac{1}{x^2}} &if &x\neq 0\\
0 & else \end{cases}
$$
is a prominent example of a function whose Taylor series ...
0
votes
1answer
32 views
Wave-Function Series?
So I was basically exploring the function:
$\displaystyle {\text{frac}(x)}$ which is the fractional part function and I noticed that it has a nice fourier series definition which is:
...
1
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0answers
77 views
Taylor series expansion example
I was reading an article and there was a snippet with a taylor series expansion as shown below:
My question is, should (11) read as $F(xA+h)+(xΔA+Δh)\frac{\partial}{\partial x}F(xA+h)$ instead of ...
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votes
2answers
103 views
Power expansion for the square root of an even degree polynomial
I am reading an article from 1936 with something that looks like an easy way to solve Riccati equations with variable coefficients as nice polynomials.The link is : ...
1
vote
1answer
203 views
Roots of truncated taylor series of exp and lambertW
If you map the nth roots of unity $z$ with the function $-W(-z/e)$ you get very close approximations to the roots of the scaled truncated taylor series of $\exp$. Here W is the lambertW function, $e$ ...
0
votes
1answer
51 views
Taylor Series Expansion with e and sin
Show that when $z\neq0$,
(a) $$\frac{e^z}{z^2}=\frac{1}{z^2}+\frac{1}{z}+\frac{1}{2!}+\frac{z}{3!}+\frac{z^2}{4!}+...$$
(b) ...
1
vote
1answer
71 views
What's the Maclaurin series for $\arcsin(x)$?
I solved the problem by using a known series: $\frac{1}{\sqrt{1-x^2}}$, but the solution I got is wrong. Also, I'm not sure what to do with the constant of integration $C$. Where is my mistake?
$$ ...
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votes
1answer
29 views
Maclauren Series and taylor polynomials
Question:
Suppose that the function $k(x)$ has a maclauren series that converges $\left(-\frac{1}{2} , \frac{1}{2}\right]$ and you are told that $|k^{(n)}(x)| \leq 10$ at all $|x| \leq ...
4
votes
3answers
138 views
Summation of a series.
I encountered this problem in Physics before i knew about a thing called Taylor Polynomials My problem was that i had to sum this series :
...
0
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0answers
16 views
Question concerning expansion of the log function.
I'll get straight to it.
$\ln(x)=\int\frac{1}{x}dx
=\int\frac{1}{1-(1-x)}dx$
And $\frac{1}{1-(1-x)}=\sum_{n=o}^{\infty}(1-x)^n$
Am I correct so far? Because on wikipedia, the series ...
1
vote
5answers
70 views
What's the Maclaurin Series of $f(x)=\frac{1}{(1-x)^2}$?
This function seemed to be pretty much straight forward, but my solution is incorrect.
I have two questions:
1. Where did I make a mistake?
2. I learned that there are shortcuts for finding a series ...
2
votes
1answer
51 views
Taylor polynomial approximation
How do you determine if adding more terms to the Taylor polynomial will improve its approximation of $f(p)$ or in other words, how do you determine if a Taylor series converges for a particular value ...
0
votes
1answer
67 views
Using the series of $\tan^{-1}(x)$ for calculating $\pi$
The power series expansion of $\tan^{-1}(x)$ is
$$\tan^{-1}(x)=x-\frac 13 x^3+\frac 15 x^5-\frac 17 x^7+ \cdots .$$
Use the above series to determine a series for calculating $\pi$.
