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

10
votes
1answer
451 views

Abel limit theorem

I would like to know if the Abel limit theorem works if the limit is infinite. Let the series $\sum_{k=0}^\infty a_k x^k$ have radius of convergence 1. Assume further that $\sum_{k=0}^\infty a_k = ...
8
votes
2answers
151 views

Convergence of differentiated power series

Consider $\displaystyle f(z)=\sum_{k=0}^\infty a_k z^k$ and suppose that $\displaystyle\sum_{n=0}^\infty f^{(n)}(0)$ converges. Prove that $\forall z \in \mathbb C, ...
4
votes
3answers
264 views

Asymptotic expansion of $J(t) = \int^{\infty}_{0}{\exp(-t(x + 4/(x+1)))}\, dx$

I want to derive an asymptotic expansion for the following Bessel function. I think I need to rewrite it in another form, from which I can integrate it by parts. I am interested in obtaining the ...
3
votes
2answers
99 views

Power series relation

Draks gave the identity, Higher Order Trigonometric Function $$\sum_{k=0}^\infty \frac{(-1)^k x^{km}}{(km)!}=\frac{1}{m}\sum_{k=0}^{m-1} \exp( e^{i\frac{2k+1}{m}\pi}x )$$ How can this be proven?
3
votes
3answers
257 views

the sum of a series

I am stuck on the computation of the following sum: $$\sum_{k=0}^{\infty} {\Big( {\frac{q}{k+1}} \Big)}^k ,$$ where $k$ is a natural number, and $0<q<1$.
2
votes
1answer
116 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} ...
2
votes
3answers
144 views

Estimating the series: $\sum_{k=0}^{\infty} \frac{k^a b^k}{k!}$

Any idea on how to estimate the following series: $$\sum_{k=0}^{\infty} \frac{k^a b^k}{k!}$$ where $a$ and $b$ are constant values. Greatly appreciate any respond.
11
votes
3answers
220 views

Could we show $1-(x-\dfrac{x^3}{3!}+\dfrac{x^5}{5!}-\dots)^2=(1-\dfrac{x^2}{2!}+\dfrac{x^4}{4!}- \dots)^2$ if we didn't know about Taylor Expansion?

Suppose that humanity haven't discovered Taylor Series Expansion of trigonometric functions or of any function that would help us on this. Which means we are not allowed to replace the given infinite ...
9
votes
3answers
526 views

Solution of $y''+xy=0$

The differential equation $y''+xy=0$ is given. Find the solution of the differential equation, using the power series method. That's what I have tried: We are looking for a solution of the form ...
9
votes
3answers
377 views

Can the the radius of convergence increase due to composition of two power series?

When composing power series, is the radius of convergence the minimum of that of the individual series, or is it like for multiplication and addition of power series where the resultant radius of ...
9
votes
4answers
643 views

The Power of Taylor Series

I am teaching a Calculus class and we are finishing up power/Taylor series this week. The last section of the chapter is on applications, but the only ones listed there are approximating non-rational ...
8
votes
4answers
221 views

Real-analytic $f(z)=f(\sqrt z) + f(-\sqrt z)$?

Are there nonconstant real-analytic functions $f(z)$ such that $$ f(z)=f(\sqrt z) + f(-\sqrt z)$$ is satisfied near the real line ? Also can such functions be entire ? And/Or can they be periodic ...
6
votes
1answer
223 views

Existence of a power series converging non-uniformly to a continuous function

I am wondering whether there exist a function $f(z) = \sum_{n\geq0} a_n z^n$ such that: $f$ converges and is continuous on the closed unit disk $D$ and the series $\sum_n a_n z^n$ does not converge ...
6
votes
3answers
504 views

Expressing the area of the image of a holomorphic function by the coefficients of its expansion

I have the following problem. Let $f:D\to \mathbb C$ be a holomorphic function, where $D=\{z:|z|\leq 1\}.$ Let $$f(z)=\sum_{n=0}^\infty c_nz^n.$$ Let $l_2(A)$ denote the Lebesgue measure of a set ...
6
votes
4answers
375 views

Power Series with the coefficients $n!/(n^n)$

I'd be grateful if someone could tell me how to obtain the convergence radius of the aforementioned power series. Or, by Cauchy Hadamard, the limit of $(n!/(n^n))^{(1/n)}$ as n approaches infinity. ...
5
votes
3answers
185 views

Series of inverses of binomial coefficients

Can you think of a simple way of proving that $$ \sum_{n=k+1}^\infty \frac{1}{n \choose k} $$ is rational for any $k \geq 2$? Here's the background. Consider a series: $$ \sum_{n=1}^\infty ...
5
votes
2answers
113 views

Sum up to number $N$ using $1,2$ and $3$

So the question asked was finding out the number of ways(combinations), a given number $N$ can be formed using the sum of $1,2$ or $3$. (eg) For $n = 8$, the answer is $10$ The given solution for ...
4
votes
1answer
124 views

Is there a generalization of the fundamental theorem of algebra for power series?

Given the similarity between polynomials and power series, I was wondering if there is any generalization of the fundamental theorem of algebra for power series. I understand that it doesn't make much ...
4
votes
3answers
508 views

What would be the radius of convergence of $\sum\limits_{n=0}^{\infty} z^{3^n}$?

I know how to find the radius of convergence of a power series $\sum\limits_{n=0}^{\infty} a_nz^n$, but how does this apply to the power series $\sum\limits_{n=0}^{\infty} z^{3^n}$? Would the ...
3
votes
1answer
393 views

Prove that if $\sum a_k z_1^k$ converges, then $\sum a_k z^k$ also converges, for $|z|<|z_1|$

Show that if a power-series converges for any value of $z_{0}$ of $z$, it will be absolutely convergent for all values of $z$ whose representation points are within a circle which passes through ...
2
votes
2answers
2k views

Radius of convergence of a sum of power series

I have two series $\displaystyle\sum_{n=1}^{\infty} a_n x^{n}$ $\displaystyle\sum_{n=1}^{\infty} b_n x^{n}$ with radius of convergence $2$ and $3$ respectively. How can I find the radius of ...
1
vote
3answers
178 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 ...
1
vote
5answers
167 views

Power series for the rational function $(1+x)^3/(1-x)^3$

Show that $$\dfrac{(1+x)^3}{(1-x)^3} =1 + \displaystyle\sum_{n=1}^{\infty} (4n^2+2)x^n$$ I tried with the partial frationaising the expression that gives me $\dfrac{-6}{(x-1)} - ...
0
votes
4answers
142 views

Find the radius of convergence of $\sum_{n=1}^{\infty}{n!x^{n!}}$

Find the radius of convergence of $\sum_{n=1}^{\infty}{n!x^{n!}}$. Should I look at this series as: $\sum_{n=1}^{\infty}({n!x^{(n-1)!})x^{n}}$? I am really confues here. In addition, any attempt to ...
0
votes
2answers
134 views

Show that $\sum_{n = 1}^{+\infty} \frac{n}{2^n} = 2$ [duplicate]

Show that $\sum_{n = 1}^{+\infty} \frac{n}{2^n} = 2$. I have no idea to solve this problem. Anyone could help me?
4
votes
0answers
233 views

Question about Big O notation for asymptotic behavior in convergent power series

Examples of such use of Big O notation can be found for instance on Wolfram Alpha here. More details on the Wikipedia page. The idea, as I understand it, is that the term between parenthesis in Big O ...
4
votes
2answers
181 views

How do I obtain the Laurent series for $f(z)=\frac 1{\cos(z^4)-1}$ about $0$?

I know that $$\cos(z^4)-1=-\frac{z^8}{2!}+\frac{z^{16}}{4!}+...$$ but how do I take the reciprocal of this series (please do not use little-o notation)? Or are there better methods to obtain the ...
4
votes
2answers
1k views

Proof of the “Radius of Convergence Theorem”

I can't figure out how it is valid to invoke the Absolute Convergence Theorem, whose hypothesis is "Let the power series have radius of convergence R", to establish case c of the Radius of Convergence ...
3
votes
2answers
78 views

Solve $(x^2 + 1)y'' - 6xy' + 10y =0$ using series method

Use series methods to solve: $(x^2 + 1)y'' - 6xy' + 10y =0$ a) Give the recursion formula b) Give the first two non-zero terms of the solution corresponding to $a_0 = 1$ and $a_1 = 0$ ...
3
votes
3answers
1k views

General term of Taylor Series of $\sin x$ centered at $\pi/4$

What is the general term for a Taylor series of $\sin(x)$ centered at $\pi/4$? It should be $(-1)^{[??]} \times \sqrt{2}/2 \times \frac{(x-\pi/4)^n}{n!}$ What power is $(-1)$ supposed to be raised ...
3
votes
1answer
2k views

Frobenius Method to solve $x(1 - x)y'' - 3xy' - y = 0$

So, Im trying to self-learn method of frobenius, and I would like to ask if someone can explain to me how can we solve the following DE about $ x = 0$ using this method. $$ x(1 - x)y'' - 3xy' - y = 0 ...
2
votes
1answer
120 views

Choice of the First Term in Legendre Polynomials

The two solutions of the Legendre's Differential Equation obtained by series solution method are : and Now according to my textbook, for the useful polynomial for n equal to a positive integer, ...
2
votes
2answers
8k views

Maclaurin Series for $\arctan(x)$ by successive differentiation

I am trying to find a Maclaurin Series for $\arctan(x)$ up to the term with the fifth power of x and I have to use the method of successive differentiation. I know (from an example in my notes) the ...
2
votes
1answer
354 views

function with isolated singularity on the unit circle and coefficients of its taylor expansion [duplicate]

Let $f$ be a function holomorphic in an open set containing the closed unit disc $D(0,1)$, except at the point $z_0$ with $|z_0|=1$, where $f$ has an isolated singularity. If $a_n$ are the ...
2
votes
1answer
159 views

What are the subsets of the unit circle that can be the points in which a power series is convergent?

Let $A\subset\Bbb C$ be a subset of the unit circle. Consider the following condition on $A$. Cond. There exists a sequence $\{a_i\}_{i=1}^\infty$ of complex numbers such that $$\sum_{n=1}^\infty ...
2
votes
3answers
102 views

If $a_0\in R$ is a unit, then $\sum_{k=0}^{\infty}a_k x^k$ is a unit in $R[[x]]$

Let $R$ a ring, and let $$\displaystyle R[[x]]=\left\{\sum_{k=0}^{\infty}a_k x^k\;\middle\vert\; a_k\in R\right\}$$ with addition and multiplication as defined for polynomials. We have that $R[[x]]$ ...
1
vote
2answers
58 views

An easy question on complex

Let $\{u_{k}\}_{k=1}^{\infty}$ be a complex number sequence. If $\sum_{k=1}^{\infty}\lambda^{k}u_{k}=0$, for each $\lambda\in \mathbb{D}(0, 1/3)$(where the $\mathbb{D}(0, 1/3)~$denotes an open disc ...
1
vote
1answer
304 views

Fibonacci Generating Function of a Complex Variable

So I'm doing work on the Fibonacci Numbers, and I came across this problem for the generating function for the recursive fibonacci numbers. I have two questions: 1. Why is it useful to use a ...
0
votes
3answers
54 views

A simple series

I don't do math a long time, so I completely don't remember how to prove that: $$ \sum_{i=1}^\infty \frac{i}{2^i} = 2 $$ Can anybody help me?
0
votes
2answers
86 views

Showing that $f(x)^3 + g(x)^3 + h(x)^3 - 3f(x)g(x)h(x) = 1$ for functions $f$, $g$, and $h$ defined by certain power series [duplicate]

I'm having trouble with this question, I have found the interval of convergence of $h(x)$ to be $(-\infty, \infty)$, but I don't know how to use that for the question as well as the hint. Any help ...
11
votes
1answer
631 views

Sum of Squares of Harmonic Numbers

Let $H_n$ be the $n^{th}$ harmonic number, $$ H_n = \sum_{i=1}^{n} \frac{1}{i} $$ Question: Calculate the following $$\sum_{j=1}^{n} H_j^2.$$ I have attempted a generating function approach but ...
12
votes
2answers
357 views

Finding the derivative of $x\uparrow\uparrow n$

I am trying to find a general derivative for the function: $f(x)=x^{x^{x^{...^{x}}}}$however to do that I must find $f^{\prime }$ and $f^{\prime \prime}$...etc. I am now trying to write down a general ...
9
votes
2answers
397 views

Origin and use of an identity of formal power series: $\det(1 - \psi T) = \exp \left(-\sum_{s=1}^{\infty} \text{Tr}(\psi^{s})T^{s}/s\right)$

The following is a historical question, but first some background: Let $\psi$ be a linear operator from a vector space to itself. The following two expressions, viewed as formal power series, can be ...
6
votes
2answers
225 views

Prove $(1-x)^{2k+1} \sum\limits_{n\ge 0}\binom{n+k-1}{k}\binom{n+k}{k} x^n = {\sum\limits_{j\ge 0} \binom{k-1}{j-1}\binom{k+1}{j} x^j} $

I stumbled upon the identity $$(1-x)^{2k+1} \sum\limits_{n\ge 0}\binom{n+k-1}{k}\binom{n+k}{k} x^n = {\sum\limits_{j\ge 0} \binom{k-1}{j-1}\binom{k+1}{j} x^j}. $$ The right-hand side is a polynomial. ...
5
votes
1answer
128 views

what is the summation of such a finite sequence?

The summation is: $$\sum_{i=0}^n \binom{2i}i \binom{2n-2i}{n-i}$$ The answer is $4^n$. How to prove it, and how to think out it?
5
votes
2answers
272 views

Basic guidance to write a mathematical article.

I'm trying to put together a mathematical article on how to obtain certain infinite series for some well known functions by a method of integrals (I like to call it "The Integral Method" - thank you), ...
3
votes
2answers
238 views

How to calculate the series?

How can we calculate the series: $$ F(x)=\sum_{n=1}^{\infty}\frac{(-1)^n}{1-x^n} $$ I found that $$ ...
10
votes
1answer
290 views

Is this generalization of an exercise in Stein true?

The following question is exercise $14$ in chapter $2$ in Stein and Shakarchi's Complex Analysis. Suppose that $f$ is holomorphic in an open set containing the closed unit disc, except for a pole ...
2
votes
1answer
122 views

Summation of exponential series [duplicate]

Evaluate the limit: $$ \lim_{n \to \infty}e^{-n}\sum_{k = 0}^n \frac{n^k}{k!} $$ It is not as easy as it seems and the answer is definitely not 1. Please help in solving it.
14
votes
2answers
442 views

Finding the convergence interval of $\sum\limits_{n=0}^{\infty} \frac{n!x^n}{n^n}$.

I want to find the convergence interval of the infinite series $\sum\limits_{n=0}^{\infty} \dfrac{n!x^n}{n^n}$. I will use the ratio test: if I call $u_n = \dfrac{n!x^n}{n^n}$, the ratio test says ...