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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1answer
24 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
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3answers
84 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
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1answer
14 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 ...
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1answer
22 views

prove that Radius of convergence is 1 [on hold]

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
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1answer
38 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
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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
17 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 $[x_0,\infty)$. ...
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0answers
26 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
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1answer
12 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 ...
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2answers
36 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 ...
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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
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3answers
325 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
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5answers
125 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 ...
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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 ...
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0answers
20 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
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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
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2answers
28 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: ...
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5answers
49 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
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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
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0answers
58 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
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0answers
29 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
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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
62 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
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1answer
23 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
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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
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1answer
33 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
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1answer
19 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
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2answers
32 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 ...
0
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1answer
29 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
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2answers
55 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
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2answers
18 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
40 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
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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
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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
146 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 ...
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3answers
108 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
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0answers
19 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
32 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
56 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} ...
1
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1answer
41 views

Finding the coefficients of $h(z)$ laurent series

Consider: $$h(z) = \frac{\psi(-z)}{(z+1)(z+2)^3}$$ Find the coefficients $a_n$ of the Laurent Series of $h(z)$ centered at $z=-2$ I got this from the approach here: Infinite sum complex analysis ...
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0answers
27 views

An application of the residue theorem

How would one find the residue of the function around an arbitrary point $z_0$ and using the branch cut $z\in(0,2\pi]$ $$R(z) = \frac{1}{(\sqrt{z+a}+\ln(z+b))^n}$$ Where $a,b\in\mathbb{R}$ and $n$ ...
0
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0answers
15 views

expansion in reciprocal terms

Given a function of the form $$T(x) = \frac{1}{(\sum_{i=1}^{\infty}c_i x^i)^n}$$ Is there ever a way to write it as the sum of reciprocal terms in $x$ .i.e a series of the form ...
1
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2answers
92 views

Is $\bigl(\sum {{x^n}\over{n!}} \bigr) \bigl(\sum {{y^n}\over{n!}} \bigr) = \bigl(\sum {{(x+y)^n}\over{n!}}\bigr)$ generalizable for series?

Before I had to do a proof demonstrating the properties of exponential multiplication using power series expansions: $$ e^xe^y=e^{x+y}, $$ and the easiest and quickest way I could think of doing this ...
0
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0answers
38 views

Lambert W function, W(x), representation for entire domain

The Taylor series for the Lambert W function is $W_0(x)=\sum_{n=1}^\infty\frac{(-n)^{n-1}x^n}{n!},\left\{x\in\mathbb{R}:|x|<\frac{1}{e}\right\}$. Is there any exact closed form way to express ...
1
vote
1answer
24 views

Prove by induction that $\sum_{\varnothing\ne S\subseteq[n]}(\prod S)^{-1}=n$.

I'm having a hard time visualizing how to prove the following by induction: For every positive integer $n$, let $[n]$ denote the set $\{1,\ldots,n\}$. Let $A$ be a set. Use the notation $P(A)$ ...
1
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1answer
42 views

Fourier Series and differential equation with epsilon

Happy New Year! I am stuck for days on expressing the solution of a differential equation using Fourier series. The question is: Consider the equation: $$\ddot{x}+x+\epsilon\left(\alpha ...
1
vote
1answer
22 views

Power series convergence in boundary problem

Say I have a power series $\sum_{k=0}^\infty a_k x^k $ which converge uniformly on $\left[0, 1\right)$ . Now I need to prove that series $\sum_{k=0}^\infty a_k $ are convergent. My idea is to use ...
2
votes
0answers
21 views

Asymptotic behaviour of $\sum_{k=0}^\infty \frac{n^k}{(k!)^\nu}$

Let $\nu>0$ be fixed. I am interested in the asymptotic behaviour of the series \begin{equation*}s(n,\nu)=\sum_{k=0}^\infty \frac{n^k}{(k!)^\nu} \end{equation*} in the limit $n\rightarrow\infty$. ...
0
votes
0answers
9 views

seeking upper/lower bounds of a function $F(m)$ related to Jacobi theta function

I am looking for the upper/lower bounds of function $F(m)$ defined and plotted above. The function is related to Jacobi theta function $\theta(x)$ and its derivative values at $x=1$: ...