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).

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0
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3answers
296 views

How do I write $e^{(-x/2)}$ as a summation?

I am new to power series. I know how to write $e^x$ as a summation, but i do not know how that helps me.
2
votes
1answer
55 views

Finding the value of $a^x$

We have the series expansion $e^x = 1 + \frac{x}{1!}+ \frac{x^2}{2!}+ \frac{x^3}{3!}+ \frac{x^4}{4!}...\infty$. Is it possible to write $a^x$ in the similar form, where ...
1
vote
0answers
63 views

Is $\sum_{n=3}^\infty \dfrac{z^n}{n \ln n}$ uniform coverge on $\lvert z\rvert \ <1$??

I tried to solve this problem using Cauchy's convergence criterion. Maybe that problem's answer is not uniform converge. But I didn't solve it. Please reply why this power series is not uniform ...
0
votes
2answers
78 views

Power series representation

I'm trying to find the series representation of $ f(x)=\int_{0}^{x} \frac{e^{t}}{1+t}dt $. I have found it using the Maclaurin series, differentiating multiple times and finding a pattern. But I think ...
1
vote
1answer
2k views

Find all the values of x such that the given series would converge.

$\sum_{n=1}^{\infty} (5^n (x-9)^n) / (n+9) $ So this is a homework question and we have unlimited tries to check our answer, however, the answer I got as well as my friend who has been helping me ...
-1
votes
1answer
154 views

Solve a differential equation using the power series method

Problem By assuming a power series solution of the form $$y(x) = \sum_{m=0}^{\infty} c_mx^m , \quad c_0 \not =0 $$ Show that the equation $ 2y'+xy=x $ has general solution ...
6
votes
2answers
63 views

Writing $f\in L^2([-\pi,\pi])$ as a power series.

Consider the space $L^2([-\pi,\pi])$. I want to show that every function $f\in L^2([-\pi,\pi])$ can be written as a power series. I remember a result that polynomials are dense in $L^2([-\pi,\pi])$. ...
8
votes
1answer
762 views

Deriving Maclaurin series for $\frac{\arcsin x}{\sqrt{1-x^2}}$.

Intrigued by this brilliant answer from Ron Gordon, I was attempting to find the Maclaurin series for $$f(x)=\frac{\arcsin x}{\sqrt{1-x^2}}=g(x)G(x)$$ with $g(x)=\frac{1}{\sqrt{1-x^2}}$ and $G(x)$ ...
0
votes
1answer
71 views

Question about a Maclaurin series of an elementary function

The Maclaurin series expansion for $(1+z)^\alpha$ is as follows: $$(1+z)^\alpha = 1 + \sum_{n=1}^\infty \binom{\alpha}{n}z^n$$ with $$|z|<1$$ What I don't understand is why is $|z|<1$?
4
votes
1answer
93 views

Approximating $e^{inx}$ by polynomials

Show that every function $e^{inx}$ can be uniformly approximated on $[-\pi,\pi]$ by polynomials in $x$. Using the power series expansion, ...
2
votes
2answers
610 views

Show that $\sum_{n=0}^\infty r^n e^{i n \theta} = \frac{1- r\cos(\theta)+i r \sin(\theta)}{1+r^2-2r\cos(\theta)}$ [closed]

Show that $$\sum_{n=0}^\infty r^n e^{i n \theta} = \frac{1- r\cos(\theta)+i r \sin(\theta)}{1+r^2-2r\cos(\theta)},$$ where $0\leq r <1$. Using this, prove that $\sum_{n=0}^\infty r^n ...
0
votes
1answer
634 views

Prove Abel's convergence theorem

In our analysis class today, our teacher wanted us to prove the following theorem, or according to him, known as Abel's theorem: If $\sum\limits_{n=0}^{\infty} a_n (z-z_0)^n$ with $a_n \in ...
4
votes
1answer
271 views

Simple Frobenius problem without recurrence relation?

I am just learning frobenius method in my 'math methods in physics' class. The first problem i am trying to solve is $$ x^2y''-xy'+n^2y=0$$ (where n is a constant). I know that i have to plug in the ...
1
vote
1answer
34 views

Convergence of “sliced” power series

Let $\phi(t)=\sum_{k=1}^\infty a_k t^k$, $x=t^m \in \mathbb{C}$ for some fixed $m\in \mathbb{N}$ be a convergent power series. I guess that $a_0=0$. For $r=0,\ldots,m-1$ and $k=mq+r$, why are the ...
3
votes
1answer
90 views

For what complex $z$ the series converges

Can someone help me with this assignment? Find for what $z \in \mathbb{C}$ the series converges $\sum_{n=1}^{\infty} \frac{(2n)!}{(n!)^2}z^n$. I've just calculated (by using Cauchy-Hadamard ...
4
votes
3answers
561 views

Radii of convergence of real analytic functions

I am proposing you a self-posed question, but I do not know whether it makes sense or not. Consider an analytic function $f\colon \mathbb R \to \mathbb R$. For every $x$ define $R(x)$ as the ...
2
votes
2answers
641 views

Laurent series for $\frac{e^z}{1 - z}$ for $|z| > 1$

I do this like $\dfrac{e^z}{1-z}$ = $-e \dfrac{e^{z-1}}{z-1}$ = $-e \sum_{k=0}^{\infty} \dfrac{(z-1)^k}{k!(z-1)}$ = $-e \sum_{k=0}^{\infty} \dfrac{(z-1)^{k-1}}{k!}$ However doesn't this give me the ...
5
votes
2answers
110 views

Laurent series of $z^{-3}$ at $z_0 = i$. Is there a way to do this by hand or is the question just evil?

I have to find the two Laurent series expansions of $\frac{1}{z^3}$ about $i$. The only approach I can think of is to do: $$\frac{1}{z^3} = \frac{1}{(z-i)^3} \left( \frac{z-i}{z} \right) ^3 = ...
2
votes
0answers
26 views

Does this function have a name? $f(z)=\sum_{k=1}^{\infty}\frac{1}{k^k} z^k$?

Let $$f(z)=\sum_{k=1}^{\infty}\frac{1}{k^k} z^k$$ This series converges absolutely for all $z \in \mathbb{C}$, by comparison with $e^z$. Is there a name for $f$? I typed it into Wolfram, but no dice. ...
1
vote
1answer
29 views

How can I conclude $R=1$?

$a_n$ be a sequence of complex number such that $\sum |a_n| <\infty$, $\sum n|a_n|=\infty$, Then I need to find the Radius of Convergence of $\sum a_nz^n$ $\lim |a_n|=0$ from the first condition ...
6
votes
2answers
232 views

Identifying a power series

I'm no analyst, so when a student in the class to whom I was teaching some elementary theory of (power) series, asked about this: ...
2
votes
3answers
195 views

Evaluate this power series

Evaluate the sum $$x+\frac{2}{3}x^3+\frac{2}{3}\cdot\frac{4}{5}x^5+\frac{2}{3}\cdot\frac{4}{5}\cdot\frac{6}{7}x^7+\dots$$ Totally no idea. I think this series may related to the $\sin x$ series ...
-1
votes
3answers
51 views

Power series $ \sum_{r=1}^{n}x^{r}=\:?$

I want to know a formula for $$\displaystyle \sum_{r=1}^{n}x^{r}=\:?$$ I can't say i can see where to derive it from at all. Any help or pointers would be greatly appreciated. Thanks
3
votes
1answer
77 views

Radius of Convergence of a Series

How would I find the radius of convergence of the following series? $$ \sum_{n=1}^{\infty}\frac{(-1)^n}{n}z^{n(n+1)} $$ The ratio test and root test are inconclusive, so I think I have to use the ...
4
votes
1answer
182 views

Maclaurin series for $e^z /\cos z$.

I want to find the Maclaurin series for the function $$f(z)=\frac{e^z}{\cos z}.$$ Right away I can tell that the radius of convergence will be $\pi/2$, since it's the distance to the nearest ...
2
votes
1answer
89 views

Converging or diverging series?

I have the following series: a) $\displaystyle \sum_{n=3}^\infty\frac{\sqrt{n}\cdot\cos^2(n)}{n^2-2}$ and b) $\displaystyle \sum_{n=1}^\infty\frac{(n+1)!(n+2)!}{n^{2n}}$ I have concluded that for a ...
2
votes
1answer
162 views

Determine the radius of convergence of $\sum_{n=0}^\infty \cos{(\alpha \sqrt{1+n^2})}z^n.$

Determine the radius of convergence of $$\sum_{n=0}^\infty \cos{(\alpha \sqrt{1+n^2})}z^n,$$ if $\alpha$ is a real constant. What about if $\alpha \in \mathbb{C}$? My attempt: at infinity, $\alpha ...
0
votes
1answer
66 views

Automata and power series

I am taking a class on Automata and Formal Languages and I need to solve an exercise, but I have no idea where to start from. It sounds like this: Decide the coefficients of the words in ...
1
vote
2answers
343 views

Find the Radius of Convergence of the Series $\sum a_{n}x^{n^{2}}$ Using $\sum a_{n}x^{n}$?

I want to show that $$\sum a_{n}x^{n^2}$$ has radius of convergence of 1, using the fact that the power series $\sum a_{n}x^{n}$ has radius of convergence $R>1$, where $R$ is a real number ...
2
votes
1answer
37 views

Are power series in a normal matrix themselves normal?

Are (convergent) power series in a normal matrix themselves normal? I have looked around for this result, and not found it. How might we prove it?
1
vote
1answer
30 views

Is there a fast way to compute coefficient of some term of the product of some series'?

The example in wikipedia is $$A=1-3x+5x^2-7x^3+9x^4-11x^5+\cdots$$ $$B=2x+4x^3+6x^5+\cdots$$ $$AB=2x-6x^2+14x^3-26x^4+44x^5+\cdots$$ And the term $x^5$ is given by ...
1
vote
1answer
99 views

Result of $\sum_{n=1}^{\infty} \frac{\sin(nx)}{n}$

I already showed that $\displaystyle g(x)=\sum_{n=1}^{\infty} \frac{\sin(nx)}{n}$ converges uniformly on $[\delta,2\pi-\delta ]$ for $\delta>0$ Now I have to calculate the limit of ...
0
votes
1answer
81 views

find the power series expansion of $f(z)$ at $z_0$ without computing derivatives

Is it possible to find the power series expansion of $f(z)=\log z$ at $z_0=1$ without computing derivatives?
2
votes
1answer
226 views

Complex power series expansion for $f(z) = \frac{e^z}{a-z}$

I have the following homework problem in my complex analysis class: Find the complex power series expansion for the function $$ f(z) = \frac{e^z}{a-z}$$ where $a \in \mathbb{C}$, and $z \ne 0$. I know ...
2
votes
1answer
67 views

Convergence of infinite series-1

To investigate the convergence/divergence of this series: $$\sum_{n=1}^{\infty} (n^{-1/2}-\sin(n^{-1/2}))^{1/2} $$ So, I took the maclaurin series$$ \sin(x)\sim x-\frac{x^3}{3!} $$ Replacing it ...
2
votes
1answer
216 views

Complex Analysis - Taylor Series

The question reads as follows: Let the function $f$ be given by $f(z)=\frac{z}{2}+\frac{z}{e^z-1}$ if $z\neq0$ and $f(z)=1$ if $z=0$. Show that $f$ is analytic at $z=0$ and that $f(z)=f(-z)$. ...
1
vote
1answer
204 views

Introduction to Analysis: Multiplication Theorem for Series

I've been stuck on this problem over the weekend so I decided to ask for some direction. The problem reads: "The multiplication theorem for series requires that the two series be absolutely ...
1
vote
1answer
65 views

Introduction to Analysis: Power Series

I've been stuck on this problem over the weekend so I decided to ask for some direction. The problem reads: "The multiplication theorem for series requires that the two series be absolutely ...
2
votes
1answer
56 views

Want to check analyticity of a series on a open disk.

How do we check the analyticity of a any power series? For example: How will we show that $$f(z):= \sum_{n=1}^\infty z^{n!}= z^1+z^2+z^6+z^{24}......+z^{n!}......$$ is anaytic on disk {$z : ...
2
votes
2answers
563 views

Radius of Convergence of $\sum ( \sin n) x^n$.

Thank you very much in advance for any assistance/advice on solving this problem. I am fairly new to power series and determining the radius of convergence. Determine, with proof, the radius of ...
8
votes
6answers
772 views

Show that if $r$ is an nth root of $1$ and $r\ne1$, then $1 + r + r^2 + … + r^{n-1} = 0$.

Show that if $r$ is an nth root of $1$ and $r\ne1$, then $1 + r + r^2 + ... + r^{n-1} = 0$. I think I can represent all the roots of 1 as follows: $r = 1^{\frac{1}{n}} ( \frac{\cos{2\pi k}}{n} + ...
5
votes
5answers
114 views

What is this limit equal to:

What is the following limit equal to and how do I prove it? $$\lim_{x\to 0^+} \frac{1}{1-\cos(x^2)}\cdot \sum_{n=4}^\infty{n^5x^n} $$ I've tried l'hospital but it doesn't seem to help since I don't ...
1
vote
1answer
64 views

Trouble finding the Laurent Expansion .

I'm having trouble progressing through (in my experience) the tedious calculations required to obtain a Laurent Expansion of a complex function. The problem arises in finding the series within the ...
2
votes
1answer
272 views

Intuition regarding Taylor series for $\frac{e^z}{1-3z}$.

The question asks me to find the Taylor series for $$f(z)=\frac{e^z}{1-3z}.$$ The radius of convergence is $|z|<1/3$ and I know the expansions for $e^z$ and $1/(1-3z)$ are \begin{align} e^z ...
3
votes
1answer
260 views

How to identify this power series as $k\sin(k/x)$?

In this question, a functional equation is solved for functions with a power series. We find a recursive formula: (copied from the answer by user achille hui) \begin{align} ( 2^1 - 3 ) a_2 &= 0\\ ...
5
votes
1answer
82 views

Why do convergence issues not play a role when talking of generating functions

This question has been on my mind for some while now. Perhaps it has a very simple answer. When we talk of generating functions we take a series $\sum a_ix^i$ and do not usually bother about it as a ...
3
votes
0answers
144 views

Given a meromorphic function (via its Laurent series), how to obtain the (Taylor series of the) two holomorphic functions it is the quotient of?

Since any meromorphic function $f:\mathbb C\to\mathbb C$ can be expressed as the quotient of two entire functions, i.e. $f(z) = \frac{n(z)}{d(z)}$ where the zeros of the denominator $d(z)$ are $f$'s ...
4
votes
1answer
98 views

Domain of convergence of power series-2

Here is the series: $$ \sum_{n=2}^\infty(-1)^n\frac{(x-3)^n}{(\sqrt[n]{n}-1)n} $$ What is the interval of convergence? I tried using root test and ratio test but finding the limit from thereon is ...
1
vote
2answers
326 views

How to denote the opposite case of the Kronecker Delta?

The Kronecker delta is defined as link to wikipedia: $$\delta_{l,m} = \begin{cases} 1 & \text{if }m=l,\\ 0 & \text{if }m\neq l. \end{cases}$$ I would like to denote the case where: $$ = ...
0
votes
1answer
201 views

Solve $2(x+1)y' = y$ using Power Series.

Given the ODE: $2(x+1)y' = y$ How can I solve that using Power Series? I started to think about it: $ \\2(x+1)\sum_{n=1}^{\infty}{nc_nx^{n-1}}-\sum_{n=0}^{\infty}{c_nx^n}=0 ...