Tagged Questions
1
vote
1answer
44 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 ...
5
votes
4answers
137 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 ...
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 ...
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 ...
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
1answer
57 views
Why do power series converge to a function symmetrically?
Why does the taylor series of $\ln (1 + x)$ only approximate it for $-1<x \le 1$?
The selected answer to the above question says that for a a power series, the interval of convergence for the ...
1
vote
3answers
32 views
Demonstrate that $\frac{1}{e^e} - 1 + e - \frac{e^2}{2} + \frac{e^3}{6}\ge 0$
How do I prove the inequality?
$$\frac{1}{e^e} - 1 + e - \frac{e^2}{2} + \frac{e^3}{6} \geq 0$$
I can see that $e^e = \sum_{k=0}^{\infty} \frac{e^k}{k!} = 1 + e + \frac{e^2}{2} + \frac{e^3}{6}+\dots ...
0
votes
2answers
46 views
Taylor series of $f(x^2)$
If you know the taylor series for $f(x)$ can you find the taylor series for $f(x^2)$ by letting $x = x^2$? The taylor series in question is $\cos(x^2)$
I know the taylor series for $\cos(x)$ is ...
1
vote
1answer
50 views
Why do Maclaurin series approximate a function for negative domain values?
A common analogy used as an intuitive explanation for a Maclaurin series is that of a car. If you know the position, velocity, acceleration, jerk etc. of a car at time zero, you are able to predict ...
2
votes
0answers
162 views
power series of arcsin(x) centered at x = 0
I am trying to prove that the Taylor expansion of $\arcsin(x) = \sum\limits_{n=0}^\infty \cfrac{(2n!)x^{2n+1}}{(2^nn!)^2(2n+1)}$.
Sorry about the notation, I'm not sure what syntax to use. S stands ...
1
vote
1answer
69 views
Adding two Power series or Maclaurin sums together and their radius of convergence
Say you have two power series. One of them has ROC of 2, and the other one has an ROC of 4. If you add the two series together is the ROC ALWAYS the lesser ROC? It seems to be a trend I've noticed, ...
0
votes
0answers
40 views
Taylors Inequality to evaluate $f(x) = x\sin(x)$ when $a = 0$ and $-1\le x\le1$
Trying to calculate the error of this function when you use a Taylor expansion to degree 4.
I keep getting $.039$ when the answer in the back of the book is $.042$.
I take the fifth derivative of ...
1
vote
2answers
32 views
Lagrange remainder to approximate $3^{2.1}$ less than 0.1
How do I solve this problem:
Use the appropriate Taylor polynomial $P_n(x,c)$ to estimate $3^{2.1}$ with error less than $0.1$, given $\ln 3$ is about $1.099$.
I understand that the remainder ...
1
vote
1answer
41 views
Find taylor polynomial that approximates e^x with accuracy at least 1.
Find Taylor polynomial at $x=0$ which approximates $e^x$ with accuracy at least $1$ for each $x \in [-2,2]$.
I dont undestand these questions that involve the $n^{th}$ remainder. I know I need to ...
2
votes
2answers
118 views
Confused by Laurent series
A typical problem related to Laurent series is this:
For the function $\frac 1{(z-1)(z-2)}$, find the Laurent series expansion in the following regions: $\\(a) |z|<1, \\ (b) 1<|z|<2, ...
2
votes
1answer
52 views
Confused over analytic functions, point convergence of power series
It is well-known that a power series sums to a function that is analytic at every point inside its circle of convergence and that conversely, if a function is analytic on an open disc then its Taylor ...
3
votes
2answers
102 views
How do I obtain the Laurent series for $f(z)=\frac 1{\cos(z^4)-1}$ about $0$?
I know that $$\cos(z^4)-1=-\frac{z^8}{2!}+\frac{z^{16}}{4!}+...$$ but how do I take the reciprocal of this series (please do not use little-o notation)? Or are there better methods to obtain the ...
2
votes
2answers
48 views
Solving limit by substituting a power series
I dont understand why I am getting 2 and the textbook says it is -2.
$$\lim_{x\to 0} \frac{1-e^x}{\sqrt{1+x}-1}$$
I subbed the power series for $e^x$ and $(1+x)^{1/2}$ then got rid of the $1$ on top ...
1
vote
3answers
48 views
Precise differences in meaning of Power Series, Taylor Series
Being an physicist/artist, not a real mathematician, I often toss around the terms "Taylor Series" and "Power Series" without any concern. Are these terms be considered interchangeable by ...
1
vote
2answers
88 views
Finding the Maclaurin series
Find the Maclaurin series for $f(x)=(x^2+4)e^{2x}$ and use it to calculate the 1000th derivative of $f(x)$ at $x=0$.
Is it possible to just find the Maclaurin series for $e^{2x}$ and then multiply it ...
1
vote
1answer
65 views
Exponential as power series
Is there a function that does not depend on $a$ such that $\sum_{x=1}^\infty \frac{a^x}{x!}f(x) = \mathrm e^{-a}$?
Just to be clear, the summation starting from 1 is intentional, otherwise the ...
3
votes
2answers
153 views
Proof of the “Radius of Convergence Theorem”
I can't figure out how it is valid to invoke the Absolute Convergence Theorem, whose hypothesis is "Let the power series have radius of convergence R", to establish case c of the Radius of Convergence ...
2
votes
1answer
49 views
Taylor series representation of a function.
I'm working on expressing the function $f(x)=\frac{6}{x}$ as a taylor series about $-4$. I've got the general idea, but I'm not quite there yet. I've come up with the equation ...
0
votes
1answer
140 views
Laurent Series and Taylor Series
I am trying to find the Laurent series of $\dfrac{1}{(1+x)^3}$; would this be the same as finding the Maclaurin series for the same function?
-2
votes
1answer
112 views
Taylor / Maclaurin series expansion origin. [closed]
Soo we all know Taylor series expansion formula for expansion around expansion point $A(a,f(a))$:
$$f(x) \approx \underbrace{f(a)}_{1st~term} + \underbrace{\frac{f'(a)\, (x-a)}{1!}}_{2nd~term}+ ...
0
votes
1answer
102 views
Laurent series of $$ g(z)=\frac{z^n+z^{-n}}{z^2-(a+\frac{1}{a})z+1}=? $$
How to find Laurent series of g(z) ?
$$ g(z)=\frac{z^n+z^{-n}}{z^2-(a+\frac{1}{a})z+1} \hspace{10mm} \begin{cases} n \in N \\
0<a<1 \end{cases} $$
answer is :
$$ ...
0
votes
1answer
91 views
Differentiating power series
Consider the power series
$$\sum_{n=0}^\infty{\frac{x^{2n}}{(2n)!}}$$
From this, it follows that its sum defines an infinitely differentiable function $f$, given by ...
0
votes
2answers
396 views
Find the Taylor Series for $f(x)$ centered at a given value $a$
$$f(x) = \frac{6}{x}\,\, \mathrm{at}\,\, a = -4 .$$
Assume that $f$ has a power series expansion. Do not show that $R_n(x) -> 0$
I took the derivatives of f(x):
$$f(x) = 6/x$$
$$f'(x) = -6/x^2$$
...
3
votes
0answers
150 views
domain of convergence of a multivariable taylor series
consider the rational function :
$$f(x,z)=\frac{z}{x^{z}-1}$$
$x\in \mathbb{R}^{+}/[0,1]\;\;$, $z\in \mathbb{C}\;\;$ .We wish to find an expansion in z that is valid for all x and z. a Bernoulli-type ...
1
vote
2answers
143 views
Why do we need Taylor polynomials?
This question doubles as "Is my understanding of what a Taylor polynomial is for, correct?" but In order to write out a Taylor polynomial for a function, which we will use to approximate said function ...
3
votes
2answers
248 views
Why is Taylor series expansion for $1/(1-x)$ valid only for $x \in (-1, 1)$?
After finding an expansion of $$ \frac{1}{1-x} = 1 + x + x^2 + x^3 + \ldots$$
a quick test of various values for $x$ reveals that this expansion is not valid for $\forall x \in \mathbb{R}-\{1\}$.
...
0
votes
1answer
115 views
Asymptotic of Taylor series
Let $f(x)$ and $g(x)$ be two Taylor series such that:
$$ f(x)= \sum_{n=0}^{\infty}(-1)^{n} a(n) x^{n} $$ and $$ g(x)=
\sum_{n=0}^{\infty} b(n) x^{n} $$, for $ a(n) >0 $ and $b(n) > 0 $.
My ...
1
vote
2answers
94 views
Problem regarding infinite sum of remainders.
Before here @math.SE there was a question regarding a problem on a maths magazine. I decided to look at the link provided, and one problem proposed was (if I'm not recalling this wrongly):
Find
...
1
vote
1answer
195 views
Power Series Definition
What does it mean for a series to be centered around a number? I'm taking complex analysis and am suddenly very confused. I didn't have this explanation, or proof of taylor and power series in ...
2
votes
4answers
153 views
Formula for calculating $\sum_{n=0}^{m}nr^n$
I want to know the general formula for $\sum_{n=0}^{m}nr^n$ for some constant r and how it is derived.
For example, when r = 2, the formula is given by:
$\sum_{n=0}^{m}n2^n = 2(m2^m - 2^m +1)$
...
0
votes
4answers
127 views
Infinite series expansion of $e^{-x}\cos(x)$
Establish an infinite series expansion for the function $y=e^{-x}\cos(x)$ from just the known series expansions of $e^x$ and $\cos(x)$. Include terms up to the sixth power.
I know that the ...
4
votes
3answers
295 views
A deceiving Taylor series
When we try to expand
$$
\begin{align}
f:&\mathbb R \to \mathbb R\\
&x \mapsto
\begin{cases}
\mathrm e^{-\large\frac 1{x^2}} &\Leftarrow x\neq 0\\
0 &\Leftarrow x=0
...
3
votes
2answers
376 views
Using Taylor series expansion as a bound
I have a function $f(x)$ that has convergent Taylor series expansion around $x=0$ in the following form:
...
2
votes
3answers
180 views
Taylor series for different points… how do they look?
I can't understand what it means to do the Taylor series at the point $a$.
The best way would be showing me how it looks for different $a$ on a graph. Do I find those graphs on the Internet?
1
vote
1answer
150 views
Maclaurin series of $\frac{1}{1+x^2}$
I'm stumped here. I''m supposed to find the Maclaurin series of $\frac1{1+x^2}$, but I'm not sure what to do. I know the general idea: find $\displaystyle\sum_{n=0}^\infty \frac{f^{(n)}(0)}{n!}x^n$. ...
0
votes
1answer
159 views
Really basic question about the Taylor expansion of a CDF
I am sorry for such a basic question... but I want to try to do a Taylor expansion on my function, which is a CDF defined over 0-1. However, when I expand around 0, which is what I read is typical, ...
2
votes
1answer
123 views
A question about the product of two series
Given two power series,
$$f(x)=\sum_{n=0}^{\infty}a_{n}x^{n}$$
and
$$g(x)=\sum_{n=0}^{\infty}b_{n}x^{n}.$$
It is easy to form their product
$$f(x)g(x)=\sum_{n=0}^{\infty}c_{n}x^{n}$$
where
...
5
votes
2answers
967 views
How to use the Lagrange's remainder to prove that log(1+x) = sum(…)?
Using Lagrange's remainder, I have to prove that:
$\log(1+x) = \sum\limits_{n=1}^\infty (-1)^{n+1} \cdot \frac{x^n}{n}, \; \forall |x| < 1$
I am not quite sure how to do this. I started with the ...
