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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2
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1answer
56 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 $|z|<1$:...
0
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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
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1answer
75 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
591 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: $$ y'' = y'(f(x)-y') $$ $$ \sum_{n=0}^{\...
2
votes
2answers
43 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: $$F_n:\Omega\to\...
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
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0answers
50 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
54 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 ...
3
votes
2answers
232 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 $...
1
vote
1answer
103 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, $$\sum_{n=0}^{\...
1
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1answer
462 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: $|z-i|&...
3
votes
2answers
101 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
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1answer
56 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
79 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 entire ...
1
vote
1answer
88 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\over(a-b)}...
1
vote
1answer
46 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
26 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 $y=\frac{x}{f(x)}$ ...
1
vote
2answers
73 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
204 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
659 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
25 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 ...
2
votes
1answer
57 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
187 views

Some inequalities for an entire function $f$ [CSIR-NET-2014]

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} |f(z)|^{...
1
vote
1answer
54 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 ...
0
votes
0answers
33 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 $n$ ...
1
vote
2answers
127 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 ...
1
vote
0answers
102 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 $W(x)$...
1
vote
1answer
31 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 $P(A)$ ...
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vote
1answer
95 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 x^2\...
1
vote
1answer
33 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
32 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 $n\rightarrow\infty$. ...
5
votes
4answers
291 views

Generating series of Catalan numbers

The Catalan numbers may be defined as follows: $C_0=1$ and $$C_{n+1}=\sum_{k=0}^n C_k C_{n-k}\, .$$ One way to compute these numbers is to introduce the generating series $f(x)=\sum_{n\geq 0} C_n x^...
1
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0answers
96 views

Borel-/Laplace-transform and $\psi$-function

I'm considering some family of functions whose coefficients of their power series occur in the columns of the following matrix A (of course thought as of infinite size) $ \qquad $ The ...
4
votes
2answers
270 views

How to calculate what this power series converges against? (double factorials)

I'm working on my physics master course homework and I'm given the following equation out of nowhere: $\displaystyle{ 1 + \sum_{n\ =\ 1}^{\infty}{z^n\left(\, 2n - 1\,\right)!! \over 2n!!} ={1 \over \,...
0
votes
2answers
38 views

Prove/disprove that $\sum_0^\infty a_nx^n = 0 \rightarrow a_n = 0 \text{ for all }n$ given $|x|<1$

For $|x|<1$, if the following statement is true, how to prove it? If not, how to disprove it? $$\sum_0^\infty a_nx^n = 0 \rightarrow a_n = 0 \text{ for all }n$$ In case $x$ takes any real value, ...
1
vote
1answer
132 views

Proof that $e^x$ can be expressed in a series of ascending powers of $x$

In a pure maths textbook I have, they prove that $e^x$ can be expressed as $1+x+\frac{x^2}{2}+\frac{x^3}{3!}+\frac{x^4}{4!}+\ldots+\frac{x^n}{n!}+\ldots$ However, before they prove this, they say they ...
0
votes
1answer
18 views

Right derivative of a power series

I have a power series $\displaystyle \sum \limits_{n \in \mathbb{N}} a_{n}x^{n}$ whose radius of convergence is equal to $4$. For all $x \in ]-4,4[$, let $f(x) = \displaystyle \sum_{n=0}^{+\infty} a_{...
1
vote
3answers
141 views

Multiplication of a complex function with essential singularity with another complex function with a pole at the same point

Im trying to proof or disprove the following claim: If $f(z)$ and $g(z)$ are holomorphic in an annulus $0 < |z − z(\beta)| < R$ and $f$ has an essential singularity at $z(\beta)$ and $g$ has a ...
2
votes
2answers
76 views

Function such that $f^{(n)}(0)=\frac{(n!)^2}{(2n+1)!}$

I was trying to solve another problem and come up with the problem if there is a function with closed form such that $$f^{(n)}(0)=\frac{(n!)^2}{(2n+1)!};(n\ge1).$$ I tried to check the condition for ...
0
votes
1answer
43 views

Sum of the series $\sum_{i=1}^n a^i i^r$

How can I find the sum of the series : $$\sum_{i=1}^n a^ii^r $$
3
votes
2answers
77 views

A simple Partial Differential Equation? I thought…

I was looking at the Partial Differential Equation involving function: $$ z(x,y)$$ $$ \frac{\partial z}{\partial x} + c \frac{\partial z}{\partial y} = 0 $$ Which fairly intuitively has a solution:...
2
votes
0answers
38 views

Finding the First Few Terms of the Sum of Two Infinite Series

Consider $$f(x)=\sum_{n=0}^{\infty}\sum_{j=0}^{n}\sum_{k=0}^{j}d_{k}c_{j-k}a_{n-j}x^{n-11}+\sum_{n=0}^{\infty}\sum_{k=0}^{n}e_{k}b_{n-k}x^{n-8}.$$ Given that I know the first few values of $d,c,a,e,b$...