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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29 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
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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) = ...
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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?
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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
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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 = ...
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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
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1answer
37 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
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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 ...
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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
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1answer
35 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 ...
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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
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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
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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
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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
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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 ...
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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?
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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
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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 $r$, $p$, ...
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2answers
48 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 ...
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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
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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 ...
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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: ...
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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 ...
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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
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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
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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
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0answers
222 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
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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 ...
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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
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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
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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
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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
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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
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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 ...
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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
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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
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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
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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 ...
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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
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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
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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 ...
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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
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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? $$ \sum \limits^{\propto }_{n=1}\frac{(-1)}{n(\ln(n+1))^{2}} $$
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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 ...
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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
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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
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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)= ...
2
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1answer
28 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
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0answers
40 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
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0answers
32 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 ...