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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1answer
20 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 ...
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0answers
12 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 $$c_j(n)=\frac{2 (2j+1)}{\Gamma(j)}\sum_{k=1}^{n}(\pi k^2)^{j}\tag{1}$$, $$P(z)=\sum_{j=1}^{m(n)}(-1)^jc_j(n)\left(z^{4j+1}+z^{-(4j+1)}\right)\tag{2}$$ ...
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0answers
52 views

Compute the sum $\sum_{i=1}^{\infty}\frac{x^ {3i}}{(3i)!}$

Compute the sum $\sum_{i=1}^{\infty}\frac{x^ {3i}}{(3i)!}$. I have no idea to find this sum. Can anyone give me a hint? Thank you in advance !
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0answers
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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
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1answer
56 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 ...
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17 views

Find the radius of convergence of the multiplication of two power series [on hold]

let there be ∑∞k=0akxk power series with radius of convergence R1, ∑∞k=0bkxk power series with radius of convergence R2 and ∑∞k=0akbkxk power series with the radius of convergence R. i need to prove ...
-1
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1answer
42 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 ...
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2answers
42 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
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1answer
47 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 $x=\pm e$. ...
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1answer
24 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 ...
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0answers
18 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
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0answers
23 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 ...
1
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1answer
53 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{ ...
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0answers
38 views

Power series expansion of $z\mapsto \frac{1}{1+z^2}$ around arbitrary point $x\in \mathbb R$

Determine the power series expansion of $z\mapsto \frac{1}{1+z^2}$ around $x\in\mathbb R$ with the respective radius of convergence. At first I tried working with Cauchy's integral formula to ...
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2answers
73 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 ...
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0answers
16 views

log - log plot. [on hold]

Show that the relation $$N(s) = \frac{a}{s^α}$$ will show a linear dependency in a log - log plot. Can you please help me about choosing values for $s$ and $N(s)$? It's related to power law.
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3answers
70 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 ...
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3answers
46 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 ...
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1answer
27 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 - ...
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0answers
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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 ...
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0answers
78 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
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1answer
55 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
52 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
24 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) ...
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0answers
29 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
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1answer
66 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 = ...
3
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1answer
21 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 ...
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2answers
20 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 $$ ...
0
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3answers
49 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
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2answers
33 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
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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 ...
1
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1answer
26 views

rewriting product of power series

According to $$\Lambda(\tau;q)=B_0(\tau)+\sum_{i=1}^\infty B_i (\tau) q^i$$ we define $$[\Lambda(\tau;q)-B_0(\tau)]^m=\bigg[ \sum_{i=1}^\infty B_i (\tau) q^i ...
0
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1answer
27 views

Reciprocal of power series with same radius

Let $f$ be a power series $f(x)=\sum a_n x^n$ with radius $R=\limsup \frac{1}{(\sqrt{|a_n|})^\frac{1}{n}}$ defined in $]-R,R[$. Let us suppose that $|f(x)|>c$ for a given $c$. Claim: Its ...
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0answers
35 views

How do I get these equations ($n$th-order deformation equation and its initial/boundary conditions)?

Earlier, I asked a question What does 'equating the like-power of $q$ ' mean? and I already got the answer about the meaning of a particular phrase in a certain context. However, my real question ...
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1answer
26 views

What does 'equating the like-power of $q$' mean?

I am reading a book "Homotopy Analysis Method in Nonlinear Differential Equations" by Shijun Liao chapter 13 Applications in Finance: American Put Options. It is stated there that Substituting ...
1
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3answers
71 views

Series $\frac{x^{3n}}{(3n)!} $ find sum using differentiation

Find sum of the series $$\sum_{n=1}^{\infty}\frac{x^{3n}}{\left(3n\right)!}$$ using differentiation. So far I found that $$S(x)+1=S'''(x)$$ but it does not help. Then I tried different interesting ...
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1answer
27 views

Problem with the inverse expansion

Let $q=e^{2\pi i z}$ and $t=q-12q^2+66q^3-220q^4+495q^5-...$ Then why is the inverse expansion equal to $q=t+12t^2+222t^3+...$? I also do not understand the notation here: $t$ means $t(z)$ or $t(q)$? ...
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0answers
30 views

Laurent series of $f(z)=\frac{1}{z(z-1)(z-2)}$

I am trying to find the Laurent series of the function $$f(z)=\frac{1}{z(z-1)(z-2)}$$in the rings: 1) $0<|z-1|<1$, 2) $1<|z-1|$, 3) $1<|z-2|<2 $ First I expressed $f$ as ...
2
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2answers
71 views
+50

radius of convergence of Taylor series, function with branch cuts

Let $f(z) $ being the analytic continuation of some holomorphic function, having many branch points and isolated singularities at $\beta_1,\beta_2,\ldots,\beta_n,\ldots$ is the radius of convergence ...
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0answers
35 views

how to prove uniform convergence of truncated product $\cos_n(z)$ to $\cos(z)$ in the strip $\Im(z)<1$?

The function $\cos(x)$ can be expressed as an infinite product in terms of its zeros $$\cos(z)=\prod_{k=0}^{\infty}\left(1-\frac{z^2}{((2k+1)\pi/2)^2}\right)\tag{1}$$ Let us define ...
2
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2answers
50 views

How do you show that a power series of $e^x$ at a point converges to $e^x$?

My task is this: Find Taylor-expansion of $e^x$ at the point $1$, the convergence interval $I$ and then show that the series converges to $e^x$. My work so far: By using $Te^x @ x= 1$ we should get ...
2
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1answer
88 views

How to find the bound of this sum?

Let $t>0,a(t)=\arg(\Gamma(1/4+it))$,$\kappa(n)=\frac{1}{2}x\pi n^2$,we need to calculate the bound,$A(x)$, of the following finite sum: $$ S(x)=\sum_{1\le n\le ...
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2answers
23 views

Designing a Power Series with certain $R$

Out of interest, is there a way to design a series with a certain radius of convergence? For example, $R=8$, or is there a way to turn a series for which the Radius of Convergence is known, then ...
0
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1answer
8 views

Radius of convergence of complex power series using Cauchy's integral formula

I have a question as follows. Let $$f(z)=\frac{\sin z}{(z-1-i)^2}$$ and $$a_n=\frac{f^{(n)}(0)}{n!}$$ Determine the radius of convergence of $$\sum_{n=0}^{\infty}a_nz^n$$ In my class we have ...
1
vote
1answer
38 views

$n$th root of power series when its coefficients are from a field with positive characteristic

Let $k$ be algebraically closed field of characteristic $p>0$. Let's consider a power series $f(x,y)\in k[[x,y]]$. Under what conditions (on $n$, $f$, ...) there exists $g(x,y)\in k[[x,y]]$ such ...
0
votes
1answer
47 views

Writing the product $\sum\limits_{r=0}^\infty \frac{z^r}{r!} \sum\limits_{s=0}^\infty \frac{z^{-s}}{s!}$ as a power series in $z$

My lecturer states that the product $$\sum_{r=0}^\infty \frac{z^r}{r!} \sum_{s=0}^\infty \frac{z^{-s}}{s!}$$ can be written as (with $n = r-s$) $$\sum_{n=0}^\infty z^n\sum_{r=n}^\infty ...
2
votes
1answer
40 views

Radius of convergence from recurrence with variable coefficients

I am solving via power series the ivp $$y'-2xy=0,\quad y(1)=2.$$ The "solution" is $$y(x)=2\left(1+2(x-1)+3(x-1)^2+\frac{10}{3}(x-1)^3+\frac{19}{6}(x-1)^4+\frac{26}{10}(x-1)^5+\cdots\right)$$ with ...
0
votes
0answers
26 views

Calculate the radius of convergence of the following power series

Let the power serie $\sum_{k\ge0}a_k(z-a)^k$ have the radius of convergence $\rho=t\in\mathbb{R^+}$, and let $p\in\mathbb{N}$. What is the radius of the following series: a) ...
2
votes
1answer
53 views

Does $\sum_{n=0}^\infty \frac{(n!)^2}{(2n)!}x^n$ converge at the endpoints of the convergence radius?

My task is this: Find the convergence radius of$$\sum_{n=0}^\infty \frac{(n!)^2}{(2n)!}x^n.$$ My work so far: By ratio test we get ...