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
votes
2answers
60 views

What is the radius of convergence of $\sum_{k = 0}^{\infty} 3^{k^2}x^{k^2}$

Where I'm at right now: $a_k = 3^{k^2}$, so we do $\frac{1}{\lim sup_{k\to\infty}(3^{k^2})^{\frac{1}{k}}}$, which is $\frac{1}{3^k}$, but that doesn't seem right. I suppose from there, I go through ...
-2
votes
1answer
64 views

Find the interval of convergence for these 3 power series

I believe I need to use the root test and ratio test. I've solved the first two, but I'm not too sure I fully understand how to do these, so was hoping someone else could work them out so I could ...
0
votes
2answers
50 views

Calculate the radius of convergence of $\sum^\infty_{k=1} \frac{(2k-1)^{2k-1}}{2^{2k}(2k)!}x^k$

I need some assistance on calculating the radius of convergence: $\sum^\infty_{k=1} \frac{(2k-1)^{2k-1}}{2^{2k}(2k)!}x^k$ I tried the quotient criteria: ...
0
votes
2answers
140 views

Complex power series : Radius of Convergence

Could anyone please suggest me how to deal with these questions (Complex Variable) : Note that all problems are in $\mathbb{C}$. 3.1 Determine the radius of convergence $\rho$ of each of the ...
3
votes
1answer
58 views

Series approximation to $\int_0^1\sqrt{\frac{2x+3}{2u^3-(2x+3)u^2+2x+1}}du$

I have figured out by graphing that, for small $x$: $$ \int_0^1\sqrt{\frac{2x+3}{2u^3-(2x+3)u^2+2x+1}}du\approx\log(1/x)+\pi/2+O(x) $$ However, I am unable to prove that this is the case. As $x\to 0$ ...
4
votes
0answers
98 views

Is there a formula for $\sum_{n=0}^{+\infty} q^{n^3}$?

When I studyied the representation of integers as sum of squares, I found that the most powerful tool is the Jacobi Triple Product, in fact this amazing identity allows us to find more useful ...
-1
votes
1answer
87 views

How to find the Number of factors, if sum of the factors are given?

A number is expressed in terms of $(2^m\times3^n)$, Find the value of $(m,n)$ if sum of all factors of a number is $124$.
1
vote
1answer
40 views

What happens to number 3?

I'm reviewing a text on Maclaurin series. This is more of an algebraic question, anyway. How do we go from here: $$ z^2e^{3z} = \sum\limits_{n=0}^\infty \frac{z^2(3z)^n}{n!}$$ to here: $$ ...
1
vote
0answers
28 views

Did i generalize this series solution correctly?

$$ f(x) = \frac{y''}{p(x)y'} + r(x) y' $$ if all functions are expressed in their power series form, then: $$ y = \sum_{n=0}^\infty a_nx^n $$ $$ p(x) = \sum_{n=0}^\infty p_n x^n $$ $$ r(x) = ...
0
votes
0answers
40 views

Question about multiplying summations with another summation inside

I have the following: $$ y = \sum_{n=0}^\infty [x^n \sum_{k=0}^\infty (k+1)a_{k+1} P_{n-k}] \sum_{n=0}^\infty x^n[s_n - \sum_{k=0}^n a_{k+1}(k+1)R_{n-k}] $$ I can easily multiply $$ ...
0
votes
1answer
39 views

For which $z \in \mathbb c$ does this series converge?

$f(z)=\sum_1^\infty \frac{(2z)^{2k}}{2k(2k-1)}$ I didn't know how to start so I just tried the ratio test. If $|\frac{a_{n+1}}{a_n}|>0$ then the series converges. $\implies$ ...
0
votes
1answer
31 views

Convergency of the power series at two points

Consider the power series $$\sum_{n=0}^{\infty}a_{n}(z+3-i)^{n}.$$ The series converges at $5i$ & diverges at $-3i$. Then which is correct ? (a) convergent at $-2+5i$ & divergent at ...
2
votes
0answers
210 views

How to solve this recurrence relation and solving the power series

Take the following recurrence relation into account: $$ a_{n+2} = \frac{1}{(n+1)(n+2)} \sum_{k=0}^n (s_k - (k+1)a_{k+1})(n-k+1)a_{n-k+1} $$ I know that: $$ s_{2m+1} = \frac{(-1)^m}{(2m+1)!} $$ and ...
2
votes
1answer
54 views

Power series of dependent and independent variables

Let $f(z,w)$ be an analytic function in two variables where $w=w(z)$ is dependent on $z$ ($z$ is the independent variable). Then $f(z,w)$ has a power series expansion centered at $(z_0,w(z_0))$ ...
1
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0answers
37 views

Proof that $\sum_{n} a_{n}$ converges if $a_{n}=O(1/n)$, $\lim_{x\uparrow 1}\sum_{n}a_{n}x^{n}$ exists?

Does anyone know of a simple proof that $\sum_{n=0}^{\infty}a_{n}$ converges whenever the real sequence $\{ a_{n} \}_{n=0}^{\infty}$ satisfies these two conditions? $a_{n}=O(1/n)$; $\lim_{x\uparrow ...
0
votes
1answer
34 views

Find the ratio and interval of convergence for $\sum_{n=1}^{\infty} \frac{n!x^n}{1\cdot 3\cdot 5\cdots (2n-1)}$

I believe this would diverge for $x\neq 0$. After using the ratio test I obtain (x)(n+1)(sum from 1 to n of (2n-1)/(2n+1)). Taking the limit as n goes to infinity the second term blows up and the ...
0
votes
1answer
74 views

If a power series converges uniformly on $\mathbb{R}$ then it must be to $0$?

Let $f(x) = \sum a_n x^n$. Let's assume that $f(x)$ has a radius $R=\infty$ and $f(x)$ converges uniformly. Now, obviously $f(0) = 0$. Meaning, $f(x)$ pointwise converging at $x=0$. Since we assumed ...
4
votes
1answer
255 views

Infinite sum of analytic function still analytic

Consider $$ f_n(x) = n e^{-n^6(x-n)^2} : \mathbb R \rightarrow \mathbb R$$ and the series $$ f(x) = \sum_{n=1}^{\infty} f_n(x). $$ Is $f$ analytic on $\mathbb R$? A function is analytic if for ...
2
votes
1answer
88 views

Proving standard properties of sine and cosine defined by their power series

Definition: We define $\displaystyle \sin x = \sum_{n=0}^{\infty}\frac{(-1)^n x^{2n+1}}{\left ( 2n+1 \right )!}, \; x \in \mathbb{R} $ and $ \displaystyle \cos x = \sum_{n=0}^{\infty}\frac{(-1)^n ...
2
votes
1answer
162 views

Sum of exponential square series

I have a infinite sum which I wonder if it will converge to a simpler function $f(r) = \Sigma_n r^{n^2} , r<1$, I also interested in case $r$ is a complex number on unity circle $r = ...
2
votes
3answers
104 views

Simplify the sum $ \sum_{k=1}^{\infty} (\frac{1}{2})^kk $

I need some help simplifying this sum: $$ \sum_{k=1}^{\infty} \left(\frac{1}{2}\right)^kk $$ I have a feeling it's some basic series thing that I'm forgetting, but I need help nonetheless.
1
vote
1answer
23 views

Radius of convergence: Why is it $\geq 1$?

Let $X$ denote a random variable with values in $\mathbb{N}_0\cup\left\{\infty\right\}$. Let $r_X$ denote the radius of convergence of $\sum_{n\in\mathbb{N}_0}\mathbb{P}(X=n)z^n$ with ...
2
votes
1answer
58 views

Counterexample for generating function?

This is Exercise 3.1.2 from Achim Klenke: »Probability Theory — A Comprehensive Course«. Exercise: Give an example for two different probability generating functions that coincide at countably ...
2
votes
2answers
56 views

sum of a given power series

Given $\sum_{n=0}^{\infty}\frac{(n-1)(n+1)}{n!}x^n$: a) Study it's pointwise and uniform convergence. b) Find the value of the sum in the interval of convergence. For a) I found that the series ...
2
votes
1answer
28 views

Prove the series converges uniformly at $[x_0, \infty)$

Let $\sum_{n=0}^\infty a_ne^{-\lambda_n x}$, where $0 < \lambda_n < \lambda_{n+1}$. It is given that the series converges at $x_0$. Prove that the series converges uniformly at $[x_0,\infty)$. ...
1
vote
0answers
39 views

Radius of power series.

Consider the formal power series in one complex variable $z$ of the form $$f(z) = \sum_{n = 0}^{\infty} c_{n} (z-a)^{n}$$ where $a,c_n\in\mathbb{C}.$ Then the radius of convergence of $f$ at ...
2
votes
1answer
47 views

Determining uniform convergence of complex power series

I'm working on some practice problems for my complex analysis course, and I'm having trouble with uniform convergence. The question asks whether the following series converges uniformly for ...
0
votes
2answers
45 views

Power series with simple recurrence relationship: $a_{n+2} = a_{n+1} - \frac{1}{4}a_n$. How to determine corresponding closed form function?

Given: $$\sum_{n = 0}^{\infty} a_nx^n = f(x)$$ where: $$a_{n+2} = a_{n+1} - \frac{1}{4}a_n$$ is the recurrence relationship for $a_2$ and above ($a_0$ and $a_1$ are also given). Is there a nice ...
0
votes
1answer
74 views

Proof $\sum{ k{ x }^{ -k }=\frac { x }{ { (x-1) }^{ 2 } } }$

As the title says, I want to prove the following: $$\sum {k{x}^{-k}=\frac{x}{{(x-1)}^{2}}}$$ But I think I am doing something wrong. I start from the following: $$\sum{x^k} = \frac{x}{1-x} \implies ...
9
votes
3answers
563 views

Proving the sum of squares of sine and cosine using the Cauchy product formula

Here are the power series of sine and cosine: $$\sin(x) = \sum_{n=0}^{\infty} (-1)^n \frac {x^{2n+1}} {(2n+1)!}$$ and $$\cos(x) = \sum_{n=0}^{\infty} (-1)^n \frac {x^{2n}} {(2n)!}$$ How can it be ...
2
votes
5answers
129 views

$e^{x} > 1$ and $0 < e^{x} < 1$

So $$\exp(x) := \sum_{n=0}^{\infty} \frac {x^n} {n!}$$ How to prove that $\exp(x) > 1$ when $x > 0$ and moreover $\exp(x) < 1$ when $x<0$ Is it possible with induction? Or must I use ...
1
vote
0answers
27 views

What can be said about the limit of a converging infinite polynomial?

Suppose we have the following polynomial of infinite order: $f(x) = a_0+a_1x+a_2x^2+...=\sum_{n=0}^{\infty}a_nx^n$ Also suppose that $f(x)$ converges to some limit $f^*$ as $x\nearrow\infty$, i.e. ...
1
vote
1answer
54 views

What Did I Do Wrong When Solving For This 2nd Order Differential Equation? (answered myself)

$$ \frac{y''}{y'}+y' = f(x) $$ I set the following to be true: $$ y = \sum_{n=0}^{\infty} a_n x^n $$ $$ f(x) = \sum_{n=0}^{\infty} b_nx^n $$ Therefore: $$ y'' = y'(f(x)-y') $$ $$ ...
2
votes
2answers
41 views

Analyticity: Uniform Limit

Problem Consider a uniformly bounded sequence over the real line: $$f_n:\mathbb{R}\to\mathbb{C}:\quad|f_n(x)|\leq L$$ Suppose they have analytic continuations with common domain: ...
1
vote
5answers
84 views

How do you call this fact about sum of powers of n-th unity root?

I often see identity $$\sum_{k=0}^{n-1}e^{\tau ika/n} = \cases {n \quad \text{ if }n | a\\0\quad \text{ otherwise}}$$ in the context of generating functions. It allows to zero out all members of ...
2
votes
2answers
34 views

show that $ 4 = \sum_{n=1}^\infty (-2)^{n+1}\frac{n+2}{n!} $

I need to show that $$ 4 = \sum_{n=1}^\infty (-2)^{n+1}\frac{n+2}{n!} $$ by considering $$ \frac d{dx}(x^2e^{-x})$$ I found that $ \frac d{dx}(x^2e^{-x}) = 2xe^{−x}−x^2e^{−x}$ What would be the ...
1
vote
0answers
64 views

What is an elegant way to express $(-1)^k$

In computation of series, a lot of times you will find a term $(-1)^k$ jutting out in an otherwise easy to remember expression. Is there some interesting way to write $(-1)^k$ that may help in ...
3
votes
0answers
48 views

Function Looks Poisson-Like: But What's the Parameter $\lambda$?

(On pause) I have $$f\left(x\right)=-x\left( x\sqrt{4-x^2}-4\arccos\left(\frac{x}{2}\right) \right)\arccos\left(\frac{x^2+d^2-1}{2dx}\right)$$ which looks a bit like the continuous version of ...
2
votes
3answers
49 views

Taylor series of $\ln(1+x)$

So let's say we want to obtain the Taylor series for $\ln(1+x)$. We know that its derivative is $\dfrac{1}{1+x}$, which has the series $\sum_{n=0}^{\infty} (-1)^nx^n$. Can we take the antiderivative ...
2
votes
2answers
222 views

Geometric Series with coin tosses

Suppose you toss a coin and observe the sequence of H’s and T’s. Let N denote the number of tosses until you see “TH” for the first time. For example, for the sequence HTTTTHHTHT, we needed N = 6 ...
1
vote
1answer
88 views

Finding the coefficient of a power series

How would I find the coefficient of: $[x^{10}]x^6(1-2x)^{-5}$ I know that I can simplify this as follows: $[x^4](1-2x)^{-5}$ and that generally the following formula would be used to solve this: ...
2
votes
0answers
36 views

Radius of convergence of $\sum k!(x+3)^k$

$\sum k!(x+3)^k$ Ok, I've tried and I'm a bit stuck... The sum is something like: $1+(x+3)+2(x+3)^2$ So $|\dfrac{x+3}{1}|<1 \Rightarrow -4<x<-2$ The answer in the book says the radius is ...
1
vote
1answer
37 views

What is the series to converge with $1/x$ from $(1,\infty)$?

I'm trying to find an alternative series of polynomials that can pssibly converge with $\frac{1}{x}$. So far I know that the taylor series for $\frac{1}{x}$ is, as should be known, ...
1
vote
1answer
402 views

Find annulus of convergence of Laurent series

Find annulus of convergence of Laurent series $\sum_{-\infty}^{\infty}2^{-n^2}(z-i)^{n^3}$ My answer: $0<|z-i|<\infty$ $\sum_{-\infty}^{\infty}2^{-n^2}(z-i)^{n^2}$ My answer: ...
3
votes
2answers
98 views

Example of Parseval's Theorem

In the textbook "Mathematics for Physics" of Stone and Goldbart the following example for an illustration of Parseval's Theorem is given: Until 2.42 I understand everything but I don't understand ...
1
vote
1answer
54 views

Name/Topological properties of the space of formal power series $\mathcal K [x]$

So, a guest lecturer introduced a concept the other day in class. Take a field $\mathcal K$ and then take the ring of formal power series on that ring, $\mathcal K[x]$. Ignoring convergence in the ...
2
votes
2answers
76 views

Singular matrix geometric sum

What is a fast way to calculate the sum $M + M^2+M^3+M^4+\cdots+M^n$, where $M$ is an $n \times n$ matrix whose cells are either $0$ or $1$? I have researched an alternative way which makes use of ...
0
votes
3answers
53 views

Existence of an analytic function under some given conditions

Which of the followings is(/are) correct? There exists an entire function $f:\mathbb C \to \mathbb C$ which takes only real values & is such that $f(0)=0$ & $f(1)=1$. There exists an ...
1
vote
1answer
83 views

Why does Partial Fractions Decomposition fail for higher degree nominator?

I can decompose $${1\over(x-a)(x-b)} = {1\over(a-b)}({1\over(x-a)}-{1\over(x-b)}) = {1\over(a-b)}({x-b-x+a\over(x-a)(x-b)}) = {a-b \over (a-b)(x-a)(x-b)}$$ and $${x\over(x-a)(x-b)} = ...
1
vote
1answer
44 views

Taylor series expansion of $e^{x+y}$ about the point $(0,1)$

My question is: what is the Taylor series expansion of $e^{x+y}$ about the point $(0,1)$? I think the standard $e^{x+y} = 1 + x+y + 1/2(x+y)^2$ ... doesn't apply here. Thanks in advance