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
15 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 ...
0
votes
0answers
16 views

How to solve this recurrence relation and solving the power series

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

What is the name of this rule? Weierstrass factorisation theorem? Newton's Identities?

Say we have roots of a polynomial $(x+2)(x+3)$ I was watching a video and it said there's a rule that we can rewrite this as $A(1-\dfrac{x}{-2})(1-\dfrac{x}{-3})$ I was wondering what the name of ...
2
votes
1answer
21 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
vote
0answers
21 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
16 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
0answers
13 views

need help solving this series [on hold]

i'm finding it difficult finding if this series converges or diverges. any help is appreciated. $\sum _{n=0}^{\infty }\left(3^{2+n}2^{1-3n}\right)$
0
votes
1answer
27 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 ...
3
votes
1answer
80 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
43 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
38 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
85 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
17 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 ...
0
votes
1answer
22 views

prove that Radius of convergence is 1 [closed]

Let's assume that $${\left\{ {{a_n}} \right\}_{n \in {\Bbb N}}}$$ is a positive sequence number. and let $$ \mathop {\lim }\limits_{k \to \infty }{A_k}=\sum_{n=0}^{k} {a_n} = \infty $$ if $$ \mathop ...
2
votes
1answer
45 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
37 views

sum of a given power series

Given $\sum_{n=0}^{\infty}\frac{(n-1)(n+1)}{n!}x^n$: a) Study it's punctual 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
19 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
30 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 $a,c_n\in\mathbb{C}.$ ...
2
votes
1answer
14 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
42 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
44 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 ...
8
votes
3answers
336 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
127 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 ...
0
votes
0answers
33 views

Closed form of a series with sinh

Is there a simple form for following function (where $a$ and $b$ are constants)? Can it be simplified to a simple form if $a>>b$? $$ u(x) = \sum _{n=0}^{\infty } \frac{ \, (-1)^n ...
1
vote
0answers
22 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
50 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
32 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
57 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
59 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
32 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
36 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
69 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
26 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
27 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
35 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
21 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
34 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
32 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
60 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
2answers
20 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
41 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
40 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
1
vote
1answer
20 views

inverse of a power series with one specific solution

I have a school assignment and for now, I don't know where to start, I have to show that there exist a surrounding $U$ of $0$ where the following is true: If $y\in U$ , the equation $y=\frac{x}{f(x)}$ ...
0
votes
2answers
56 views

Upper Bound for $|f^{n}(0)|$ given that $f$ is Analytic

Let $f(x)$ be an analytic function in some neighborhood of $x=0$. $f$ being analytic implies that its has a convergent Taylor series expansion about $x=0$. That is, there exists $R>0$ (radius of ...
3
votes
1answer
149 views

Sufficient condition for an infinite series to be zero

Consider an infinite power series $f(x):= \Sigma_{i=0}^\infty a_n x^n$ where $a_n$ are any complex numbers. In particular, we make no assumption on $a_n$ to ensure the series converge in any ...
1
vote
3answers
109 views

Prove sum of $\cos(\pi/11)+\cos(3\pi/11)+…+\cos(9\pi/11)=1/2$ using Euler's formula

Prove that $$\cos(\pi/11)+\cos(3\pi/11)+\cos(5\pi/11)+\cos(7\pi/11)+\cos(9\pi/11)=1/2$$ using Euler's formula. Everything I tried has failed so far. Here is one thing I tried, but obviously didn't ...
2
votes
0answers
20 views

How to compute a radius of convergence?

Suppose that i have a power series defined by $$\begin{align} f(z)&=\sum_{n=0}^{+\infty}a_nz^n\\ a_0&=0\\ a_1&=1\\ a_n&=\frac{a_{n-1}+a_{n-2}+a_{n-1}a_{n-2}}{3} \end{align}$$ How i can ...
1
vote
1answer
33 views

Formal power series manipulations and a closed formula for $\sum_{n\geq 0}{\frac{n^2+4n+5}{n!}}$

I'm reading a book on generating functions, and in their formal power series section they define: If $f \overset{ops}{\leftrightarrow} \left \{ a_n \right \}_{0}^{\infty}$, and $P$ is a polynomial, ...
2
votes
1answer
57 views

Some inequalities for an entire function $f$

Let, $f(z)=\sum_{n=0}^{\infty}a_{n}z^{n}$ be an entire function and let $r$ be a positive real number. Then, which is(/are) correct? (a) $\sum_{n=0}^{\infty}|a_{n}|^{2}r^{2n}\le sup_{|z|=r} ...