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.
-2
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0answers
28 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 ...
1
vote
0answers
41 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
143 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
votes
1answer
268 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
$$
\begin{align}
...
2
votes
3answers
37 views
Finding the Taylor series of $f(z)=\frac{1}{z}$ around $z=z_{0}$
I was asked the following (homework) question:
For each (different but constant) $z_{0}\in G:=\{z\in\mathbb{C}:\,
z\neq0$} find a power series $\sum_{n=0}^{\infty}a_{n}(z-z_{0})^{n}$
whose sum ...
0
votes
2answers
27 views
Series expansion with remaining $log n$
I'm studying the asymptotic behavior $(n \rightarrow \infty)$ of the following formula, where $k$ is a given constant.
$$ \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
132 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
vote
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
995 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
votes
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
vote
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
69 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 ...
1
vote
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
204 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
vote
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
50 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
vote
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
46 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}
$$
is a prominent example of a function whose Taylor series ...
0
votes
1answer
32 views
Wave-Function Series?
So I was basically exploring the function:
$\displaystyle {\text{frac}(x)}$ which is the fractional part function and I noticed that it has a nice fourier series definition which is:
...
1
vote
0answers
77 views
Taylor series expansion example
I was reading an article and there was a snippet with a taylor series expansion as shown below:
My question is, should (11) read as $F(xA+h)+(xΔA+Δh)\frac{\partial}{\partial x}F(xA+h)$ instead of ...
0
votes
2answers
99 views
Power expansion for the square root of an even degree polynomial
I am reading an article from 1936 with something that looks like an easy way to solve Riccati equations with variable coefficients as nice polynomials.The link is : ...
1
vote
1answer
202 views
Roots of truncated taylor series of exp and lambertW
If you map the nth roots of unity $z$ with the function $-W(-z/e)$ you get very close approximations to the roots of the scaled truncated taylor series of $\exp$. Here W is the lambertW function, $e$ ...
0
votes
1answer
47 views
Taylor Series Expansion with e and sin
Show that when $z\neq0$,
(a) $$\frac{e^z}{z^2}=\frac{1}{z^2}+\frac{1}{z}+\frac{1}{2!}+\frac{z}{3!}+\frac{z^2}{4!}+...$$
(b) ...
1
vote
1answer
70 views
What's the Maclaurin series for $\arcsin(x)$?
I solved the problem by using a known series: $\frac{1}{\sqrt{1-x^2}}$, but the solution I got is wrong. Also, I'm not sure what to do with the constant of integration $C$. Where is my mistake?
$$ ...
0
votes
1answer
28 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
votes
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
66 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 $\pi$.
0
votes
2answers
44 views
Show using the series for $\cos x$, that $\cos 0$ converges to $1$
Show using the series for $\cos x$, that $\cos 0$ converges to $1$.
3
votes
1answer
73 views
Taylor expansion of an integral
I am interested in the Taylor series expansion around $t=0$ of the following expression:
$$I(t)=\int_{0}^{\infty}e^{-x^2}\log\left(e^{-(x-t)^2}+e^{-(x+t)^2}\right)dx$$
Normally, I would proceed by ...
2
votes
2answers
46 views
Analytic functions of a real variable which do not extend to an open complex neighborhood
Do such functions exist? If not, is it appropriate to think of real analytic functions as "slices" of holomorphic functions?
2
votes
3answers
62 views
Find Maclaurin series for$f(x) = \frac{2x}{1-5x^3}$
I'm trying to find the Maclaurin series for $f(x) = \frac{2x}{1-5x^3}$, but my solution is different from what I know it supposed to be, which is $2x+10x^4+50x^7+250x^{10}+...$
This is my attempt:
...
0
votes
0answers
50 views
Doubt on Taylor's expansion
If we have a $f: \mathbb{R} \times \mathbb{R}^n \times \mathbb{R} \to \mathbb{R}$ , $f= f(x,\mathbf{y},z)$, which is $\mathcal{C}^2$ with respect to $x$ and $\mathbf{y}$ for every $z$ and is ...
