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

2
votes
1answer
44 views

Function that Represents Divergent Power Series?

Suppose we have the following power series $$\sum_{k=0}^\infty\left(x^2+1\right)^{2k}$$ If we wished to find the function that represents this series, it seems reasonable to suppose that the ...
1
vote
0answers
60 views

How do I evaluate this sum :$\sum_{n=0}^\infty z^{n^3}$ and Is there a visual proof for it?

if $$\sum_{n=0}^\infty z^n = \frac1{1-z}, \quad z \in \mathbb{C},\; |z| < 1 .$$ then is there a way to deduce this sum:$$\sum_{n=0}^\infty z^{n^3}$$ from the above Identitie or any visual proof ...
1
vote
2answers
166 views

How was the equation re-written?

This question is a part of inhomogeneous recurrence relations (IHR). The actual question was Find a solution to $a_n - a_{n-1} = 3(n-1)$ where $n \ge 1$ and $a_0 = 2$. While going through the ...
1
vote
1answer
29 views

Question about the Cauchy Product and how it is done

Lets say we have the following: $$ \sum_{k=0}^\infty z^k \sum_{j=0}^k \frac{1}{j!(k-j)!} B_{k-j}^f(x) \frac{d^{j}}{dx^{j}}[a_k(x)] $$ Would it be correct to say that: $$ \sum_{k=0}^\infty z^k ...
1
vote
1answer
42 views

Find series power of $F(x) =e^{-x}x^{2}$

i need help for this problem; find a power series for $F(x) \text{=}e^{-x}x^{2} $ and derivate and prove this expression $$ \sum \limits^{\infty }_{n=1}\frac{(-2)^{n+1}(n+2)}{n!} =\text{4}$$
0
votes
1answer
21 views

Convergence radius

I know the Cauchy Hadamard equation to calculate the convergence radius of a power series $$\sum_{n=0}^{\infty} a_n x^n$$ Is there a way to generalize this for series of the form ...
0
votes
0answers
17 views

Product and Quotient of series

Is it possible to find a single power series product (and quotient) representation with same convergence interval of two functions? How are the general terms combined?
0
votes
0answers
16 views

Comparing the supremum of Maclaurian series with the function.

Suppose $f$ is an entire funciton with the Maclaurin Series $$a_0+a_1z+a_2z^2+\cdots $$ Show that if $r>0$ then $$|a_0|^2+|a_1|^2|r|^2+|a_2|^2|r|^4+|a_3|^2|r|^6+\cdots < \sup_{|z|=r} |f(z)|^2 ...
4
votes
0answers
42 views

How to compute this series?

I am stuck in computing this series (i.e, finding a closed-form formula): $$ \sum_{i=0}^k \binom{k}{i} \frac{2r^{i+1}(1-r)^{k-i+1}p^{k-i}v^i s^k}{(1-r)p^{k-i}s^i + r v^i s^{k-i}}, $$ where $r$, $p$, ...
0
votes
2answers
50 views

Monotone convergence of functions ant theor asymptotic power series

consider a sequence of functions $f_n:(0,\infty)\rightarrow\mathbb{R}$ which are positive and monotone, i.e. $$0< f_1\leq f_2\leq....\leq f_n\leq f_{n+1}...$$ Now let us assume we know the ...
0
votes
3answers
69 views

Do their exist power series with non circular regions of convergence?

So far just about any series of the form $$ \sum_{i=0}^{\infty} \left(a_ix^i \right)$$ Has tended to have a circular disk of convergence (of some radius, sometimes even 0). Is there a reason this ...
3
votes
2answers
68 views

Calculating the radius of convergence of a series.

Let $d_n$ denote the number of divisors of $n^{50}$ then determine the radius of convergence of the series $\sum\limits_{n=1}^{\infty}d_nx^n$. So obviously we need to calculate the limit of ...
1
vote
1answer
42 views

Checking uniform convergence of $\sum\frac{\left(x\ln x\right)^{n}}{n}$

Find the set where the series $\sum\frac{\left(x\ln x\right)^{n}}{n}$ converges and determine if convergence is uniform on that set. I used root test to find the set of convergence: ...
0
votes
1answer
34 views

find the sum of the following series using Maclaurins expansion

Find the sum of the following series: $$\sum_{n=0}^\infty {x^{n}}{\sinh(5n+5)}$$ The sum for $ {\sinh(5n+5)}$ is as it follows $$\sum_{n=0}^\infty \frac{(5n+5)^{2n+1}}{(2n+1)!}$$ And now I do not ...
0
votes
1answer
26 views

Maclaurin series for the function: $f(z)=\frac{1}{2+4z}$

I want to find a Maclaurin series for the function: $$f(z)=\frac{1}{2+4z}$$ and to find its radius of convergence. Now my attempt gave me:$$\sum_{n=0}^\infty ...
0
votes
1answer
18 views

Answer verification: Power series expansion of $\frac{1}{3-z}$ and radius of convergence about $3i$

Find a power-series expansion of the function $f(z)=\frac{1}{3-z}$ about the point $3i$ and calculate the radius of convergence, my attempt: $$f(z)=\frac{1}{3}\left(\frac{1}{1-(\frac ...
2
votes
2answers
48 views

What is the power series expansion for Riemann-Zeta at $0$?

What are the first few terms of the Laurent series expansion of $\zeta(0)$? It gets mentioned here but they only show the first term and I am kind of confused on how they got $-1/2$.
5
votes
0answers
238 views

Formula for composition of formal power series with binomial coefficient

Let $f=\sum\limits_{n\geq 0}{f_n x^n}$ and $g=\sum\limits_{n\geq 1}{g_n x^n}$ be formal power series. The $x^n$ coefficient of $f(g)$ is $$ \sum\limits_{\mathbb{i} \in \mathcal{C}_{n}} {f_k ...
0
votes
3answers
75 views

$\sum\limits_{n=4}^{\infty } \frac{2^n + 8^n}{10^n} = ?$

im looking for hints on how to do: $\sum\limits_{n=4}^{n= \infty } \frac{2^n + 8^n}{10^n} = ?$ I thought this may have had something to do with geometric series but nothing obvious comes up ...
1
vote
3answers
24 views

Analytic function and absolute convergence

(H. Priestley , Introduction to Complex analysis, exercise 5.5) Suppose f(z)= $\sum_{n=0}^\infty c_n z^n$ for z in $\Bbb C$. Prove that for all R: $$\sum_{n=0}^\infty |c_n| R^n \le 2M(2R)$$ where ...
1
vote
1answer
25 views

Power series expansion answer different

Power series expansion of $$f(z)=\frac{1}{3-z}$$ around $4i$. I calculated the radius of convergence to be $5$, and I obtained the power series: $$\sum \limits_{n=0}^\infty ...
3
votes
1answer
69 views

Existence of solution as a power series on non-empty interval

Consider the following differential equation: $$ x''(t) = a_1(t)\, x'(t)+a_0(t)\, x(t) $$ With $a_0,a_1$ functions defined for every real number and continuous on $\mathbb{R}$. Question: Does there ...
1
vote
0answers
29 views

Power expansion with Big O notation regarding to logarithmic.

I want to know power series expansion calculation using Big O notation. That is $$1-{\displaystyle \frac{x\log^2 (x)}{(x+1)\log^2 (x+1)}}$$ at infinity. Someone calculate easily by using Big O ...
2
votes
1answer
51 views

Series of reciprocals of a quadratic polynomial

Inspired by this question I was wondering if there is a systematic way to calculate this types of series, so my question is: Is there a general approach to evaluate (i.e., find a closed formula) ...
1
vote
3answers
48 views

Puiseux Series?

WolframAlpha says that $$\sqrt{x^2-1}$$ expanded in Puiseux series near 1 is $\sqrt 2 \sqrt{x-1}$ I don't know what is the Puiseux series, I have search on the net but I don't have understood so ...
0
votes
0answers
36 views

Power series at another point

How are the coefficients of power series of the same function at two different points related? The case I have in mind is: $$ \frac{x}{e^x-1}=\sum_{k=0}^\infty \frac{B_k}{k!} x^k, $$ where $B_k$ are ...
1
vote
1answer
33 views

The convergence of the power series $\sum \limits^{\infty }_{n=1}a_{n}(x-2)^{n}$ for various $x$

I would ask for help on how to solve this problem more specifically to know how to test whether a given $x$ converges in a power series. I would appreciate your insights. Of the power series $\sum ...
1
vote
1answer
54 views

Absolute convergence of $\sum a_n$

I would ask a help for the following problem If someone could tell me what criteria or applies so I would appreciate. Show that if $ \sum \limits^{\infty }_{n=1}a_n $ is absolutely convergent, then $ ...
5
votes
1answer
51 views

A question regarding power series expansion of an entire function [duplicate]

Let $f$ be an entire function and let for each $a\in \mathbb R$, there exists at least one coefficient $c_n$ in $f(z)=\sum\limits_{n=0}^{\infty}c_n(z-a)^n$, which is zero. Then $f^{(n)}(0)=0$ for ...
1
vote
2answers
31 views

Prove Alternating Series Approximation

Prove if $S=\sum_{n=1}^{\infty}a_{n}$ is an alternating series with $\left | a_{n+1}\right | < \left | a_{n} \right |$, and $\lim_{n\to\infty}a_{n}=0$, then $\left |S-(a_{1}+a_{2}+\cdots+a_{n}) ...
2
votes
2answers
63 views

Convergence of $\frac{1}{(\ln n)^{\ln n}}$

Could I have a hint for testing the convergence of the following series please? $$\sum_{n=2}^\infty\frac{1}{(\ln n)^{\ln n}}$$ I am very appreciative for your help.
1
vote
2answers
50 views

Is there a power series expansion of the Signum function?

I would like to define a linear operator as the sign of a another operator. But to use it I would need to expand it out as a power series. Roughly, I'm wondering if something like this ...
0
votes
1answer
16 views

Functional equation, inverse w.r.t composition, which function gives identity?

Given a function $$f:\mathbb{C}\to\mathbb{C},\qquad z\mapsto \frac{1-2z-\sqrt{1-4z}}{2z}\text{,}$$ I want to know a function $g:\mathbb{C}\to\mathbb{C}$, for which holds $$\left(f\circ ...
0
votes
2answers
26 views

Prove absolute convergence from alternants

He failed to get the show in its entirety in this series, one I could indicate how working with this kind of series? $$ \sum \limits^{\propto }_{n=1}\frac{(-1)}{n(\ln(n+1))^{2}} $$
0
votes
1answer
55 views

Prove that $\,\displaystyle f(z) = \sum_{n\ge1}\frac{z^n}{n^2}$ is univalent in the disk $\,D\big(\frac23\big)$

I'm having some difficulty with this question: Prove that the function $\,\,\displaystyle f(z) = \sum_{n=1}^\infty\frac{z^n}{n^2}\,$ is univalent in the disk $D\big(\frac23\big)$. There is the ...
1
vote
1answer
19 views

Interval of convergence? (Relatively simple question)

What is the interval of convergence of the power series: $\dfrac{(-1)^{(n-1)}x^n}{n^3}$ I know it should be |x| < 1, but does that mean the interval of convergence is $(1,-1)$ or $(-1,1]$ or ...
0
votes
1answer
25 views

$z \cdot \cot(z)$ series

Let us consider an expansion $z \cot(z) = \sum_{n=0}^{\infty}{(-4)^{n} \cdot B_{2n} \cdot \frac{z^{2n}}{(2n)!}}$. How to prove the RHS? I see possible to come to the expansion $\pi \cot(\pi z) = ...
1
vote
2answers
28 views

Power Series of a Holomorphic Function determined by its Real Part and $f(0)$?

While looking at exercise sheets from last year, I encountered the following statement but wasn't able to prove it myself. Let $f: D_R(0) \rightarrow \mathbb{C}$ be holomorphic and $ f(z)= ...
2
votes
1answer
31 views

Formula for $q$-expansion of weight 2 modular forms

Is there a general formula for finding the $q$-expansion of weight 2 modular forms?
3
votes
0answers
43 views

How to show that two probability generating functions are equal?

From Grimmett's Probability and Random Processes: Let $G_a(s) := \sum_0^\infty a_is^i$ where $a = \{a_i : i \geq 0\}$ is a real sequence. Uniqueness. If $G_a(s) = G_b(s)$ for $|s| < R'$ ...
0
votes
0answers
37 views

Question about the coefficient of operator

Note that the "coefficient of" operator is an operator that takes the coefficient of the power series. We start with the following: $$ \frac{1}{f(x)+z} - \frac{1}{f(x)} = \sum_{k=0}^\infty ...
3
votes
3answers
44 views

Power Series Coefficients

Find the sum of the coefficients of $x^{20}$ and $x^{21}$ in the power series expansion of $\frac 1{(1-x^3)^4}$. I don't know a lot on power series at the moment, and I was wondering how do I find ...
0
votes
1answer
32 views

generating function for a power sequence

The question is short: I don't understand how should I solve this. Problem wants the G(x) of this: 1,4,9,16,... I can solve this one but I cannot connect these two to each other: 1,2,3,4,...
0
votes
1answer
46 views

Complex Number question [Cauchy Integral/Series]

I'm going through the practice finals that my professor uploaded on his site, and I came across this question, and I have absolutely no one clue how to approach it and never seen anything like this on ...
2
votes
2answers
35 views

How to do power series expansion

What is the coefficient of $x^{11}$ in the power series expansion of $\frac 1{1-x-x^4}$? How do I do power series expansions?
3
votes
2answers
55 views

Is this a power series?

Is the following a power series? $$\sum_{n=0}^\infty a_k \left( \frac{2x}{1+x^2} \right)^k \ , x \in (-1,1)$$ where $a_k$ is a bounded sequence. I was asked to show that this power series converges, ...
2
votes
3answers
68 views

What is the technical difference between a formal and informal power series?

In my lecture notes the professor wrote that $$e^x = \Sigma \frac{x^k}{k!}$$ is a formal power series because we can plug in whatever we want in $x$ and both side will equate This is an obvious ...
0
votes
2answers
35 views

Condition for the convergence of a particular power series in $ℂ$

The problem is given as: Show that there exists no power series $f(z)=\sum_{n=0}^{\infty}C_nz^n$ such that:$f(z)=1$ for $z=\frac12,\frac13,\frac14,...$ and $f'(0)>0$ My approach so far: Let's ...
2
votes
2answers
46 views

Showing that $\sin'(x)=\cos(x)$

I want to show the "simple" relation: $$\sin' x=\cos x$$ by using power series. I know that: $$\sin x=\sum_{n=0}^{\infty}(-1)^n \frac{x^{2n+1}}{(2n+1)!}$$ $$\cos x=\sum_{n=0}^{\infty}(-1)^n ...
0
votes
0answers
27 views

Series identity for cotangent

How to prove that $x \cot(x) = 1 - 2 \sum_{n=0}^{\infty}{\frac{x^{2}}{(n \pi)^{2}-x^{2}}}$? First, it does not seem to be solvable, using considerations regarding Taylor series. The Fourier approach ...