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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0answers
16 views

How to find the power series of the inverse of a function?

Question: I wish to find the inverse of the following function: $f(x)=\frac{1}{2}\left(\arctan(x) + \ln\left(\sqrt{\frac{1+x}{1-x}}\right)\right)$ This is the equation for a radial null geodesic in a ...
3
votes
1answer
41 views

Limit of given expression

Let $\sum a_k=s$. I want to show that $$\lim\limits_{x\to 1^-}(1-x)\sum\limits_{k=1}^{\infty}\frac{ka_kx^k}{1-x^k}=s$$ where $x\in(0,1)$. Thanks for your helps.
2
votes
1answer
34 views

converging power series over $p$-adic integers is a UFD

Denote by $|\cdot |$ the $p$-adic norm, and let $$\mathbb Z_p \{z\}=\left\{\sum_{n=0}^\infty a_nz^n;\ a_n\in \mathbb Z_p ,\ |a_n|\underset {n\rightarrow \infty} {\longrightarrow 0} \right\}$$ the ...
1
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0answers
32 views

How to manipulate the bound on the summation

$$ B_n^{f^2}(x) = \sum_{k=1}^n\sum_{j=0}^{n-k} 2^{k-j} {j+k \choose j} \frac{d^j}{df^j}[f^k] B_{n,j+k}^f(x) $$ I am looking to have the bounds switched, can someone show me exactly how this is done? ...
2
votes
2answers
45 views

Why is the integral starts from $0$?

Consider $$f(x) = \sum_{n=0}^\infty \frac{(-1)^n}{3n+1} x^{3n+1}$$ It's a power series with a radius, $R=1$. at $x=1$ it converges. Hence, by Abel's thorem: $$\lim_{x\to 1^-} f(x) = ...
-1
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2answers
60 views

Evaluate the sum below [on hold]

Evaluate the following sum $$1*1!+2*2!+3*3!+....+1000*1000!$$ any help guys?
-3
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2answers
26 views

Prove Sum Approximation Theorem [on hold]

Prove if $S=\sum_{n=0}^{\infty}a_{n}x^{n}$ converges for $|x|<1$, and if $|a_{n+1}|<|a_{n}|$ for $n>N$, then $$|S-\sum_{n=0}^{N}a_{n}x^{n}|<|a_{N+1}x^{N+1}|/(1-|x|)$$ I have already proven ...
1
vote
0answers
26 views

Sum of gamma-ish power series

I'm wondering if there is a nice closed-form expression for the sum $$ \sum_{n=0}^{\infty} n^{-\alpha} x^n, \quad \alpha \in (1,2), \; x \in (0,1) $$ This is a power series with coefficients $a_n = ...
0
votes
4answers
67 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 ...
2
votes
1answer
38 views

Uniform convergence in the endpoints of an interval

Study the pointwise and uniform convergence of the series $$\sum_{n=1}^\infty\dfrac{4^n}{n^2}\dfrac1{(1+x^2)^n}$$ I'm doing this exercise and I'm not sure about the following: What I've done ...
1
vote
1answer
28 views

How do I find the interval of convergence?

Suppose I have: $$\sum \cfrac{(-1)^n}{\sqrt{n}}x^n$$ If I use the ratio test, I get $$\cfrac{1}{\sqrt{1+\frac{1}{n}}}|x|$$ Why can it be said the radius of convergence here is $1$? Disregard the ...
0
votes
2answers
51 views

Closed form of series

writing these in closed forms. Firstly i want to know if there is a specific way to solve these and if yes what should i look at before i approach these problems. ...
2
votes
1answer
36 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
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0answers
56 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
162 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
26 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
41 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
15 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
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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
49 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
67 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
65 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
40 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
30 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
16 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
46 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
223 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
71 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
22 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
23 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
68 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
27 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
40 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
36 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
32 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
52 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
45 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
29 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
44 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
15 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
25 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
53 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
21 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
27 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)= ...