Questions about the properties of functions of the form $\sum_{n=0}^{\infty}a_n x^n$, where the $a_n$ are real or complex numbers, and $x$ is real or complex (or more generally an element of a Banach algebra).

learn more… | top users | synonyms

0
votes
1answer
118 views

Multiplicative Inverse of a Power Series

For a formal power series $$F(x) = \sum p_i x^i$$ a multiplicative inverse of $F$ exists iff $p_0 \neq 0$. The inverse $\sum q_i x^i$ satisfies the recursion $$q_0 =\frac{1}{p_0}\\ q_{n} = ...
2
votes
0answers
39 views

Summing the series $\sum_{k = -\infty}^{\infty} e^{ja_{0}^kt}$

For a fixed $t$, I wish to find the value of, $S = \sum_{k = -\infty}^{\infty} e^{ja_{0}^kt}$ where it is known that $a_{0} > 1$. I tried to express $S$ as a $z$ transform of some known function, ...
0
votes
1answer
90 views

Find the radius of convergence of the power series

$\displaystyle\sum_{n=0}^{\infty}a_nz^n$, where $a_{2k+1} = 2^k$ and $a_{2k} = (1 + (1/k))^2$ for $k = 0, 1, 2, \dotsc$ I started off by doing the ratio test, but I know that the ration test is for ...
0
votes
3answers
60 views

How do I simplify the formal power series $1+2x^2F(x)^2$?

I have this formal power series $$F(x)=1+2x^2F(x)^2$$ that I want to put into non-recursive form. I can expand, $$1+2x^2F(x)^2=1+2x^2(1+2x^2F(x)^2)^2= 1+2x^2+8x^4F(x)^2+8x^6F(x)^6$$ and I could ...
2
votes
1answer
38 views

Power series centered at $x =0$

I have this question in my advanced calculus textbook. Give an example of a power series centered at $x = 0$ that converges on $[-2,2]$ but not absolutely on the entire interval $[-2,2]$, and ...
1
vote
0answers
17 views

When is a series expansion related to its derivative by a polynomial equation?

Is there some common theory behind the following two examples? Example 1. Let $p(t) = \sum_{n \geq 0} (-1)^k t^{2k}/(2k)!$, and $x = p(t), y = p'(t)$. Then $x^2 + y^2 = 1$ identically. Example 2. ...
2
votes
0answers
25 views

Finding a base for a series to sum to a constant

I'd like to find the value of $r$ that solves the following equation: $$\sum_{n=1}^N r^{\frac{-1}{n}} = C \,,$$ where $N$ and $C$ are positive constants. An approximate method would also work fine ...
8
votes
2answers
392 views

Taylor Series of $\frac{1}{1-\cos x}$

The problem is, as the title suggests, to find the Power Series Expansion of $\frac{1}{1- \cos x}$ around $x=c$. What I've tried: Direct Computation: Derivatives get very ugly quickly, and don't ...
0
votes
2answers
41 views

Question on generating power series for a function

When generating power series for the function $y = 1/(2-x)$ I can see two different ways of solving this question, but with very different answers. SOLUTION: $$y=\frac{1}{1-(x-1)} = ...
1
vote
1answer
78 views

Radius and Interval of Convergence for Power Series

Find the radius and interval of convergence for the power series $\displaystyle{\sum_{k=1}^{\infty}} \frac{(x+3)^k}{k(6+(-1)^k)^k}$ I found that R=1 by calculating $\frac{1}{R} = ...
2
votes
0answers
352 views

What does the convergence of a Dirichlet series tells us about the convergence of a power series?

If $D(s)=\displaystyle \sum_{k\geqslant 1} f(k)\, k^{s}$ converges for $\Re(s)\lt a$, what is the radius of convergence of $\displaystyle \sum_{k\geqslant 1}f(k)\, x^k$ $=T(x)$? Conversely, what ...
0
votes
1answer
50 views

Term wise differention

Consider $S(x) = \displaystyle{\sum_{k=1}^{\infty}} x^k k^2$. (a) Find an explicit formula for $S(x)$ on the interval $-1<x<1$ by repeated termwise differentiation of a geometric series. Be ...
1
vote
1answer
32 views

Finding the Power Series of a Complex fuction.

Find a power series expression $\sum_{n=0}^\infty A_n z^n $ for $ \frac{1}{z^2-\sqrt2 z +2} $ I'm completely stuck on this question. I know how to manipulate power series but I've never had to find ...
9
votes
4answers
239 views

Closed form of $\sum \frac{x^n}{n^n}$

Is there a closed form of this series? $$ f(x) = \sum_{n=1}^\infty \frac{x^n}{n^n} $$ I tried few standard tricks how to sum a power series but none of them helped.
1
vote
1answer
91 views

Solving differential equations using power series

I need to solve this differential equation by power series: $$y''+3xy'+(2x^{2}+6)y=0$$ Any help is great! Thanks!
1
vote
1answer
43 views

Why does the only maximal of $k[[X_1,\ldots,X_n]]$ is $(X_1,\ldots,X_n)$?

I'm trying to understand in this book why the only maximal of $k[[X_1,\ldots,X_n]]$ ($k$ field) is $(X_1,\ldots,X_n)$: If I prove $rad(k[[X_1,\ldots,X_n]])\subset (X_1,\ldots,X_n)$, (where $rad$ is ...
0
votes
2answers
52 views

Find $\sum_{n=0}^\infty\frac{(a|x|)^n}{\frac{n}{2}!}$ where $\frac{n}{2}!=\Gamma(\frac{n}{2}+1)$

Find A=$\sum_{n=0}^\infty\frac{(a|x|)^n}{\frac{n}{2}!}$ where $\frac{n}{2}!=\Gamma(\frac{n}{2}+1)$. I know that A converges (I used the ratio test) but I can't work out what it converges to. ...
2
votes
1answer
24 views

Substitution of complex power series, example of my book

I'm reading this example of my book. It is in dutch, but I think it is clear even if you don't understand the words. There are two things I don't understand: Why does the sum begin with $n=1$ ...
1
vote
1answer
89 views

Is the completion of $(k[x,y]/f)_\mathfrak{m}$ isomorphic to $k[[x]][y]/f$?

Let $k$ be an algebraically closed field and let $f\in k[x,y]$ be an irreducible polynomial with no constant term that is not a polynomial in $x$ alone. Is it the case that the completion of the ...
2
votes
2answers
54 views

Difficulties evaluating the endpoints of the radius of convergence for a particular power series.

I am having difficulties evaluating the endpoints of the radius of convergence for the following power series. $$\sum_{k=0}^{\infty}\frac{(k!)^2 x^k}{(2k)!}$$ Using the ration test we get |x|<4. ...
0
votes
1answer
37 views

Approximation of $\sqrt{1+wi}$

How can $\sqrt{1+wi}$ be approximated? where $-\infty<w<\infty$; My aim here is getting rid of the square root. I've tried binomial, Maclurin and Taylor series around various points. but they ...
0
votes
1answer
74 views

please solve this diffrential equation question on power series

In the differential equation $y'' + (x-3)y' + y=0 $ of power series at $x_0=2$ , I took $ y=\sum_{n=0}^{\infty}a_n(x-x_0)^n $ ,then I tried to solve this but not getting the answer. if someone solve ...
0
votes
1answer
46 views

What is the series expansion of $f(z)\cdot\exp\left({s\,\log(z)}\right)$?

For analytic $f$, how can I represent the expression $f(z)\cdot\exp\left({s\,\log(z)}\right)$, i.e. $f(z)\cdot z^s$ in the form $$\sum_{n}^\infty\left(\sum_{k}^\infty a_k s^k\right)z^n,$$ at least ...
1
vote
0answers
100 views

Convergence set of power series

I am trying to find the convergence set of the power series: $\sum_{n=1}^\infty ln\big[1+\big(\dfrac{1}{n}\big)\big](x+2)^n$. So using the ratio test: $\lim_{n\to\infty} \dfrac{|a_{n+1}|}{|a_n|} = ...
2
votes
1answer
77 views

Radius of convergence proof

Let $a,c_0,c_1,...\in \mathbb{R}$ with at least one of $c_0,c_1,c_2,...$ nonzero, and let the power series $\sum_{n=0}^{\infty}c_n(x-a)^n$ have positive radius of convergence $r$. Show that there ...
1
vote
1answer
54 views

Question on $2^N$th Roots of Unity within a function.

Prove that, if $w$ is a $(2^N)$th root of unity, where $N \in \mathbb N$, then: $$\lim_{r\to 1^-}|f'(rw)| = \infty$$ Where: $$f(z) = \sum\limits_{j = 1}^\infty 2^{-j}z^{2^j}$$ I haven't done left ...
0
votes
2answers
50 views

Find the radius of convergence of the series

Find the radius of convergence of $$\sum_{n=0}^\infty\frac{n!z^{2n}}{(1+n^2)^n} .$$ Actual trouble is finding the limit applying Ratio test. Please help.
0
votes
3answers
144 views

How to find the sum of this power series $\sum\limits_{n=0}^\infty \frac {x^{5n}} {(5n)!}$

How to prove that $$ \sum\limits_{n=0}^\infty \frac {x^{5n}} {(5n)!}= \frac{2}{5} e^{-\cos \left( 1/5\,\pi \right) x}\cos \left( \sin \left( 1/5\,\pi \right) x \right) +\frac{2}{5}\, e^{\cos ...
3
votes
2answers
176 views

If it converges, how to show that power series converges to $f(x)$?

I had a very basic question. Suppose $f(x)$ is a function. And let us say it has a power series :- $$f(x) = \sum_{n=0}^\infty a_nx^n.$$ Suppose we are operating inside the region of convergence. ...
0
votes
1answer
44 views

A power approximation function

I am trying to construct a function that would approximate $a^b$ using Maclaurin series. Here are my reasoning: Since $$a^b=e^{b\ln a}$$ and $$e^x=\sum^{\infty}_{k=0} \frac{x^k}{k!}$$ it should ...
2
votes
0answers
122 views

How can the following “funny identity” be generalised?

When asked for a "funny identity", Andrey Rekalo answered the following: $$\left(\sum\limits_{k=1}^n k\right)^2=\sum\limits_{k=1}^nk^3 .$$ Not only do I think it's funny, I also think it's very ...
0
votes
2answers
111 views

Prove that $f(z) = \sum\limits_{k = 1}^\infty \frac{z^{2^k}}{2^k}$ is continuous in the closed unit disc and holomorphic inside it.

I have started off by assuming that there is a disc of radius $r$ for which $|z|<r$ for $r \in (0,1)$ and $z \in D_r$. This implies that $|z|^{2^k} < r^{2^k}$ And after that, I don't know ...
1
vote
1answer
54 views

Why $(1+1/2^p)^q = (1+1/2^q)^p$ implies $(p=q)$?

How can I prove that: $(1+1/2^p)^q = (1+1/2^q)^p$ (real $p\leq q$) implies $p=q$ ? Seems quite simple, but I don't understand where to start from... Thanx!
1
vote
1answer
134 views

Is there a power series representation of $\frac{1}{\zeta(s)}$?

I've seen power series representations for $\zeta(s)$, $\textit{e.g.}$ \begin{align*} \zeta(s)=\frac{1}{s-1}+\sum_{k=1}^{\infty}\frac{(-1)^{k}}{k!}\gamma_{k}(s-1)^{k} \end{align*} where $\gamma_{k}$ ...
5
votes
3answers
282 views

Maclaurin Series of $\frac{1}{e^x -1}$

I want to find the Maclaurin Series for the function $f(x) = \frac{1}{e^x -1}$. But when I compute the first derivative of f(x): $$ \frac{d}{dx}\frac{1}{e^x -1} = -\frac{e^x}{(e^x-1)^2} $$ A the ...
0
votes
1answer
41 views

Help with Taylor series problem

I am using maple to plot the graphs of e^e^x versus its truncated Taylor series around 0. For small values of x, the two graphs converge nicely, but once x<-3, my Taylor series loses control. Here ...
0
votes
1answer
40 views

Logarithm expansion

I have a problem showing that the following identity for power series holds true: $$ \ln(1+x+x^2+x^3+...)=\sum_{n\geq1}\frac{x^n}{n} $$ when $\left|x\right|<1$. Can anyone help me, please? ...
0
votes
1answer
40 views

Formal power series different from normal power series

As far as I understand formal power series is an infinite sequence unlike normal power series which try to approximate the function in a few terms. So will formalPowerSeries of sin(x) differ from ...
2
votes
0answers
35 views

A Problem about Infinite Series

There is no idea to solve the question for me. Let $T\subset\mathbb N_{>0}$ be a finite set of positive integers. For each integer $n>0$, define $a_n$ to be the number of all finite ...
-1
votes
1answer
82 views

Convergence, Divergence and Summability of this series

If f(x) is an infinitely differentiable function at x=0 and $f^{(n)}(0)$ is the nth derivative of the function f at zero, then does the series below converge or diverge? $\sum_{n=0}^{\infty} ...
1
vote
1answer
144 views

Taylor series of $\frac 1 {1+x^2}$

I have to construct the Taylor series of $$\frac 1 {1+x^2}$$ around $0$ and $1$ and analyze the convergence in both cases. Also (but this is a consequence of the previous series) I have to construct ...
3
votes
2answers
92 views

Show that $\sum_{n=1}^{\infty}z^{n!}$ diverges for infinitely many $z$ with $|z|=1$

Problem: I need to show that the power series $\sum_{n=1}^{\infty}z^{n!}$ diverges for infinitely many $z$ with $|z|=1$. I tried to prove it by contradiction by assuming that diverges for finitely ...
0
votes
3answers
54 views

A simple series

I don't do math a long time, so I completely don't remember how to prove that: $$ \sum_{i=1}^\infty \frac{i}{2^i} = 2 $$ Can anybody help me?
5
votes
2answers
342 views

Express $(1-z)^{-1}$ as a power series around $z_0=-1+i$.

I need to express $(1-z)^{-1}$ as a power series in powers of $(z+1-i)$. I would like some guidance on the complex analogue of power series and in writing out this particular case. Many thanks for ...
0
votes
3answers
67 views

How to get power by knowing the number and result

How to get power by knowing the number and result. For Example $$2^n = 8$$ how can i return the power $n$ by knowing number $2$ and result $8$ or $$4^n = 1024$$ how can i return the ...
0
votes
1answer
66 views

Does $\sum_{n=0}^\infty\frac{a^n}{\frac{n}{2}!}x^n$ converge?

And if so, what is the radius of convergence of $x$? I am inclined to think it converges absolutely for all $x$ but I can't prove it. I have tried using an adaptation of the ratio test: ...
1
vote
1answer
37 views

A problem about power series and big-O

The problem is: Prove: There exist constants $a$, $b$ such that $\frac{z^3-5z^2+3z}{(z+2)^3}=1+\frac{a}{z}+\frac{b}{z^2}+O(\frac{1}{z^3})$ as $z\rightarrow \infty$ and find an explicit values for $a$ ...
2
votes
2answers
223 views

Evaluation morphisms of formal power series and nilpotent elements

Given a commutative ring $A$, and a finitely presented (associative) $A$-algebra $B$, show that a morphism of $A$-algebras $A[[x]] \longrightarrow B$ is given by evaluation at an nilpotent element $ ...
7
votes
1answer
193 views

The series $2+3x+5x^2+7x^3+11x^4+…$

It occurred to me to ask whether the power series whose coefficients are the primes has non-zero radius of convergence, and if so, what kind of function it produces. Wikipedia has some bounds on the ...
1
vote
1answer
110 views

Radius of Convergence of $\sum\ z^{n!}$

Does anyone know how to find the radius of convergence of the series $\sum\ z^{n!}$, where $z$ is a complex number? I tried to use the definition: $\frac{1}{R}=Limsup|\frac{a_n+1}{a_n}|$, but I ...