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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3
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3answers
457 views

Interesting Series with Zeta Function

I was trying to find another representation for the value of an integral when I found the following series: $$f (z)=\sum_{n \in \Bbb N} (-z)^{n-1}\frac {(2^n-1)}{2^n}\zeta (n+1) $$ For $|z|<1$ and ...
0
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1answer
30 views

Creating Formula from Data Series

I have a power-law-ish data series and need to back my way into a formula for it, such that it can be generated for any number of points. This is the n=20 version (value rounded to nearest 0.25). ...
1
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1answer
44 views

Geometric proof of expansions of series

I have read that Barrow had proved the fundamental theorem of calculus. I have read that proof and its a good. Further I know Newton had derived the sine and cosine series. His methods obviously didn'...
0
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1answer
34 views

Logarithm on the reals: negative power series?

I am well aware that $ln(x)$ has no Laurent series on the complex plane, because of its multi-valuedness, but I have always wondered whether a similar beast would exist if we restricted our attention ...
2
votes
0answers
45 views

Radius of convergence of power series $\sum\limits_{n=1}^{\infty}n!x^{n!}$ (different methods yield different results)

I have to find the radius of convergence of $\sum\limits_{n=1}^{\infty}n!x^{n!}$. It is a power series, therefore: One of the ways to find the radius of convergence is to find $\lim \dfrac{|n!|}{|(...
1
vote
1answer
44 views

Taylor series for $\tanh(z)$

Find the Taylor series of $\tanh(z)$ around $z_0=0$. $z$ is a complex variable. I can use all the basic series as facts like the $\cosh$ and $\sinh$ series. I know how to calculate the series ...
1
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0answers
28 views

Convergence of $\sum_{n=1}^{\infty}\log(1-e^{2\pi inz})$

Let $f(z)=\sum_{n=1}^{\infty}\log(1-e^{2\pi inz})$ be given on the upper half plane $H=\{z\in\mathbb{C}:\Im(z)>0\}$. Why does this function converges absolutely and uniformly on compact subsets of $...
0
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0answers
24 views

Branch points of power series and radius of convergence

The following text trys to give an answer to the following problem: Let $t(q)=\sum_{n=0}^{\infty}t_n q^n$ be a powerseries with radius of convergence $1$ and $t_0=0,~t_1\neq 0$. $w(q)$ is another ...
1
vote
1answer
16 views

Convergence of factor

In my math course there are some examples to test convergence of power-series with d'alembert. One of the examples is : $\sum_{n=0}^\infty \frac{x^n}{n!}$ Now i tried to solve this with d'...
1
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1answer
34 views

Unclear definition concerning convergence of a power series

I am following a course on computer algebra and at the end of the course, my professor wrote something down I could not follow at first. Concretely it handles about a definition given as follows: ...
1
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1answer
22 views

Convergence of sum of power series and numerical series.

Considering the following series $$\sum_{n=1}^{\infty} \frac{n}{2^n}(x+2)^n + \sum_{n=1}^{\infty} \frac{n^3}{\sqrt {n!}}$$ We need to calculate the domain of convergence of this series. Well the ...
1
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1answer
26 views

Radius of convergence

What is radius of convergence of $\sum_{n=2}^{\infty} \left(1+\dfrac{1}{n}\right)^{n^2}z^n$? I don't know anything
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2answers
37 views

Expanding a function into a power series

I was given this function $$\frac{x-2}{1-x}$$ around x zero = 2 I'm not sure how to do it, i called x-2 = t, so 1-x = -t-1 and i moved on from there.. Is there another way to do this question?
1
vote
1answer
42 views

Calculus 2 - Prove Disprove - convergence of Taylor series

I got this question regarding properties of Taylor series. I'm stuck on the second question, I believe it is true since the area of convergence for X is affected by the coefficient and it is not ...
1
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1answer
14 views

About the set of $x$ values at which the Taylor series of $f(x)$ converges to $f(x)$

Let $f(x)$ be a function (for simplicity, let us assume that it is defined on $\mathbb{R}$ and infinitely differentiable), and $T$ the Taylor series of $f$ at $x=a$, with interval of convergence $I$. ...
6
votes
0answers
108 views

Help with the following summation when $x^{37}=1,x\neq 1$

I want to find the following summation $\text{Let }x^{37} = 1 \text{ and } x \neq 1,$ $\\ \text{Find the summation of }$ $$\frac{1}{(1+x+x^2+x^3)^3}+\frac{2}{(1+x^2+x^4+x^6)^3}+...+\frac{36}{(1+x^{...
1
vote
2answers
93 views

would changing the lower limit of a power series affect radius of convergence

When we change the lower limit of a power series by any finite quantity, would it increase or decrease radius of convergence or no change? Clarification of terminology: There might be confusion about ...
1
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1answer
36 views

Radius of convergence of $\sum_{k=0}^{\infty} c_n^2x^n$.

The problem statement is as follows: Suppose the radius of convergence of the complex valued series $\sum_{n=0}^{\infty} c_nz^n$ is $R$. Find the radius of convergence of $\sum_{n=0}^{\infty} c_n^2z^n$...
0
votes
0answers
55 views

Is it possible to find $g(\kappa)$ in this equation

I have ran into the following integral equation as part of my research. For $\xi = (\alpha\theta)^{1/\alpha}$ and for all $\theta>0$. I have the following equality $$\int\limits_0^\infty g(\...
-4
votes
2answers
114 views

Identifying $\sum\limits_{k=0}^\infty\frac{1}{k!}\frac{x^{k+6}}{k+6}$ [closed]

I'm trying to prove this equality. $$\sum_{k=0}^\infty\frac{1}{k!}\frac{x^{k+6}}{(k+6)} {=} e^x(x^5-5x^4+20x^3-60x^2+120x-120)+120$$ posted by: http://math.stackexchange.com/q/832368. How do I get ...
1
vote
1answer
14 views

Help to prove an expression about sums of binomials coefficients using Complex Power Series theorem.

I'm solving some exercises from Kreszig's Advanced Math book and I got stuck in one: (10th ed, chapter 15.3, problem 18): Using $(1+z)^p*(1+z)^q=(1+z)^{p+q}$, obtain the basic relation: $$\sum_{n=0}^...
0
votes
2answers
31 views

Radius of convergence of a series which diverges when every term is made positive

$\{\,a_n\mid n \ge 1\,\}$ be sequence of real numbers. Partial sum of $a_n$ forms a convergent series. Partial sum of absolute value of $a_n$ forms a divergent series. Let radius of convergence of ...
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1answer
23 views

Series Solution For ODE

I am currently working on some introductory problems for series solutions for ODEs and am really struggling. The question is as follows: $$ (7+x)y' = y $$ Calculate the first five terms in the series....
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0answers
23 views

How to prove a self-recirpocal polynomials $P(z)$ to have all its zeros on the unit circle $|z|=1$?

Let $m(n)=10(n+1)^3$ and $$c_j(n)=\frac{2 (2j+1)}{\Gamma(j)}\sum_{k=1}^{n}(\pi k^2)^{j}\tag{1}$$, $$P(z)=\sum_{j=1}^{m(n)}(-1)^jc_j(n)\left(z^{4j+1}+z^{-(4j+1)}\right)\tag{2}$$ $$Q(z)=z^{4m(n)+1}P(z)=...
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0answers
31 views

An analogue of the Cauchy formula for radius of convergence for power series with arbitrary (non-integer) exponents

By Cauchy formula, the radius of convergence of the series $\sum_{n=0}^{\infty}a_nr^{n}$ is $\rho=1/\limsup\limits_{n\rightarrow +\infty}\sqrt[n]{|a_n|}$. Let $\{\lambda_n\}_{n=0}^{\infty}$ be an ...
3
votes
1answer
61 views

For which values of $a\in\mathbb{C}$ does $\sum\limits_{n=1}^\infty\frac{a^n}{n}$ converge?

For which complex values of $a$ does $\sum\limits_{n=1}^\infty\frac{a^n}{n}$ converge? Clearly when $|a|>1$ it does not and when $|a|<1$ it does, so we only have to see what happens when $|a|=...
-1
votes
1answer
49 views

The $n=0$ term in the power series $\sum_{n=0}^\infty a_n x^n$

This question is about the definition and notation for the $0^{th}$ term of the power series: $$\sum_{n=0}^{\infty} a_n x^n$$ There are two possible ways to interpret this term: 1) It is just a ...
0
votes
2answers
50 views

Explicit expression of a given power series

Let us have a look to the power series of the form $$\sum_{n=0}^{\infty}{\frac{1}{n+2}x^n},\ \ \ x\in\mathbb{R}$$ I want to find an explicit expression of this power series. I think one have to us ...
2
votes
1answer
51 views

Convergence of $\sum_{n=1}^\infty \frac{n!}{n^n} x^n$

I'm trying find out where $\sum_{n=1}^\infty \frac{n!}{n^n} x^n$ converges. First I found that the radius of convergence is $R=e$, but after that I had difficulty testing convergence at $x=\pm e$. I'...
0
votes
1answer
27 views

Radius of Convergence on Power Series Help

I am struggling to find the radii of convergence of the following two series: $$\sum_{n}n^{\cos(n)}z^n$$ $$\sum_{n}(2^{-n} + 3^{-n})z^n$$ Here I tried using ratio test and lim sup, but didn't ...
0
votes
0answers
21 views

Definition of an inverse-powerseries

Let $t(q)=\sum_{n=0}^{\infty}t_n q^n$ be a complex powerseries convergent for all $|q|<1$. Assume $t_0=0$ and $t_1\neq0$. Not it says Let $q(t)$ be the local inverse of $t(q)$ with $q(0)=0$. ...
2
votes
0answers
31 views

Power series expansion of $z\mapsto \frac{\mathrm e^{z}}{1-tz}$ and $z\mapsto \tan z$

Determine the power series expansion and radius of convergence of $z\mapsto \frac{\mathrm e^{z}}{1-tz}$ around $0$ with $t\in\mathbb C$. Determine the radius of convergence and the first three non-...
1
vote
1answer
58 views

Use Power Series to solve system of differential equations

Problem: Hello, I wonder how you would use a Power Series to solve a system of differential equations. Lets say I have the system $$\begin{cases}(1)\text{ }\text{ }x_1'=2x_1+4x_2 \\ (2)\text{ }\...
1
vote
2answers
107 views

If $\sum_{n=0}^{\infty}a_{n}x^n $ converges for $|x| < R$ , then $\sum_{n=0}^{\infty}na_{n}x^n $ converges for $|x| < R$

I have the following statement: If $\sum_{n=0}^{\infty}a_{n}x^n $ converges for $|x| < R$ , then $\sum_{n=0}^{\infty}na_{n}x^n $ converges for $|x| < R$ as well. I couldn't find a ...
0
votes
3answers
76 views

Evaluate $\int_0^1y ( ( 1+\frac{1}{y^2} )\log (1+y^2) -1 )dy=-1+\frac{\pi^2}{24}+\log 2$ and a related generalization

Let $0<x<1$ and $0<y<1$ thus $\xi=xy^2<1$ and we can use the series expansion $$\frac{1}{2}\log\frac{1+\xi}{1-\xi}=\sum_{n=0}^\infty\frac{\xi^{2n+1}}{2n+1}$$ to get $$\frac{1}{2}\int_0^...
1
vote
3answers
52 views

Expand a function to power series

I have the following function and i try to expand it to a power series - $$F(x) = \frac{2x}{(x^2+1)^2}$$ around $X = 0$ I tried to substitute $t = -x^2$ and got stuck. I would like to get some help ...
0
votes
1answer
30 views

Find limit using Maclaurin power series

I encountered the following problem: $$ \lim_{x\to 0} \frac{x-\ln(1+x)}{x-\arctan x} $$ I expanded $ \arctan x $ in the denominator up to the fifth term and get the following: $$ x - \left(x - \...
0
votes
0answers
20 views

Change of variables in Power series

Hello StackExchange community, this is probably a dumb question, but suppose you have a function that have the following power series $f= \sum_{i = 0}^{\infty} f_i r^i$. where $f:\mathbb{R} \...
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0answers
28 views

How can the integral of the sum of a geometric series apply for r=1

Say you have the series $R(x)=\sum_{n=0}^{\infty}\frac{(-1)^n x^{2n+1}}{2n+1}$, which is convergent for $x\in[-1,1]$ Then you differentiate: $R'(x)=\sum_{n=0}^{\infty}-(x^2)^n$ This is a geometric ...
3
votes
0answers
102 views

On a family of polynomials related to the expansion of $(1+\epsilon)^{x/\epsilon}$ as a series in $\epsilon$

Consider the sequence of polynomials $(P_n)_{n\geqslant0}$ uniquely defined by the recursion $$(P_n)'=\sum_{k=0}^{n-1}\frac{P_k}{n-k+1},$$ valid for every $n\geqslant0$, with the initial conditions $...
5
votes
1answer
56 views

Asymptotic expansion of $(1+\epsilon)^{s/\epsilon}$

I have taken the logarithm of this expression and computed the Taylor expansion of the $\log(1+\epsilon)$ term but by doing this we're required to calculate powers of this series when using the ...
0
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1answer
56 views

Sophomores dream

On wiki there is a proof of Sophomore's dream. I am trying to understand what they did when changing the variable and how they got $e^{-u}$.
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1answer
28 views

Taylor series integration

I am having trouble with the following question: Integrate the Taylor series $$e^{(-t^2)} = \sum^\infty_{n=0} \frac{(-t^2)^n}{n!}$$ term-by-term to obtain the Taylor series for erf (error function) ...
0
votes
0answers
32 views

Power series representation with Gamma function

This is taken from Stein and Shakarchi's Complex Analysis (Chapter 6, Exercise 4): Prove that if we take $$f(z) = \frac{1}{(1-z)^\alpha}$$ for $|z|<1$ (defined in terms of the principal branch ...
2
votes
1answer
73 views

Laurent series of $\frac{e^z}{z^2+1}$

I cant figure out the laurent series of the following function. Let $f(z)= \frac{e^z}{z^2+1} $ and $|z|\gt 1$ $$\frac{1}{z^2+1}=\sum_{n=0}^{\infty}(-1)^nz^{-2n-2}$$ and $$e^z = \sum_{n=0}^{\infty}...
3
votes
1answer
26 views

Theorem 3.44 in Baby Rudin: Can we replace the coefficients with their absolute values?

Here's Theorem 3.44 in the book Principles of Mathematical Analysis by Walter Rudin, third edition: Suppose the radius of convergence of $\sum c_n z^n $ is $1$, and suppose $c_0 \geq c_1 \geq c_2 ...
0
votes
2answers
24 views

How to find radius of convergence with power series from differential equations

So I have a question that says find the radius of convergence after I have found the power series solution of a given differential equation. I know to find the radius of convergence you take $$ p=\...
0
votes
3answers
51 views

How do I express $f(z)= \frac{6z}{z^2 - 4z + 13}$ as a power series centered at 0?

I am having trouble solving this power series problem because I usually go about decomposing the $f(z)$ and then using geometric series, but the method doesn't seem to work with this because I get $\...
1
vote
2answers
34 views

What's wrong with my radius of convergence test?

Given $\sum \limits _{n=2} ^\infty \frac{(\ln x)^n} n$, find its radius of convergence $R$. Using the ratio test, I arrived at $$|\ln x| < 1 \implies e^{|\ln x|} < e^x \implies |x| < e \...
3
votes
1answer
29 views

One to one map $f$ equal to its power series

Across a difficult exercise sheet I encountered this exercise : Let $f$ be a continuous map from $\bar D$ the closed unit disk (in $\mathbb{C}$) to $\mathbb{C}$. We suppose that $f$ is one to one ...