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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2answers
16 views

I need help finding the interval of convergence for a power series

I need to find the interval of convergence for the following power series: I used the ratio test and got (2X)lim(n/n+1) and this is pretty much where I'm stuck.
2
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1answer
31 views

radius of convergence of a complex power series

Can you tell me what you think about my solution to this problem? In case it's wrong, or needs changes, just something like "try looking at this", "consider that"... a hint is enough, please no ...
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1answer
33 views

Find the radius of convergence of power series

Suppose that $\sum_{k = 0}^\infty a_kx^k$ has radius of convergence of $R \in (0,\infty)$. a) Find the radius of convergence of $\sum_{k = 0}^\infty a_kx^{2k}$ b) Find the radius of convergence of ...
0
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2answers
39 views

Why can we assume a certain term is $0$ in this case?

So, my differential equations book tries to guide us through the process of solving Airy's equation. We start off with $$y''-xy=0\space,\space -\infty<x<\infty$$ The book has already ...
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1answer
58 views

how to approximate this expression $\frac{1}{8}x^2(1-\frac{1}{12}x^2)/(1-\frac{1}{4}x^2)$

when x is small, for example <1, then the expression can be approximate by (from a book) $$ g(x)= \frac{-x^2}{8}{\frac { \left( 1-1/12\,{x}^{2} \right) }{1-1/4\,{x}^{2}} }= \frac{-x^2}{8} ...
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0answers
20 views

Find an infinite power series of the form an$z^n$ with radius of convergence 1 that converges for z such that |z|=1 except when z = z1, z2, …zm.

Find an infinite power series of the form $\sum_n a_n z^n$ with radius of convergence 1 that converges for every $z$ such that $|z|=1$ except when $z = z_1, z_2, \ldots, z_m$ where $z_1$, $z_2,\ldots, ...
2
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0answers
24 views

Proving the Bessel function solves the Bessel equation

Using the notation for the Bessel function as $J_n(z)=\sum \limits_{k=0}^{\infty}\frac{(-1)^kz^{n+2k}}{k!(n+k)!2^{n+2k}}$, I want to show that $w=J_n(z)$ satisfies ...
-2
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2answers
33 views

Find the radius of convergence of the power series $\sum_{n=1}^\infty x^n/{ (4-{1\over{n}})^n}$

Find the radius of convergence of the following power series $$\sum_{n=1}^\infty{x^n\over(4-{1\over{n}})^n}$$
3
votes
3answers
223 views

How to find Laurent series Expansion

$f(z)$ is defined like this: $$ f(z) = \frac{z}{(z-1)(z-3)} $$ I need to find a series for $f(z)$ that involves positive and negative powers of $(z-1)$, which converges to $f(z)$ when $0 \leq |z - 1| ...
2
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1answer
20 views

Convergence radius of a primitive of a complex power series expansion

For a complex power series expansion $$ f(z)=\sum_{n=0}^{\infty} c_n(z-a)^n $$ with convergence radius $r$, we have that for $|z-a|<r$: $$ f'(z)=\sum_{n=1}^{\infty} nc_n (z-a)^{n-1}$$ (this is a ...
1
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0answers
38 views

How to manipulate this summation in the easiest way possible?

$$ D = \sum_{k=c}^{n}\sum_{j=0}^{k-c}[{k-c \choose j}\ln^{k-c-j}(g(x))[\ln(g) f'(x) f_c^{(j)} X_{n,k(f\rightarrow g)^c} + f_{c}^{(j)} X_{n,k(f \rightarrow g)^{c}}' + \frac{d}{dx}[f_c^{(j)}] X_{n,k(f ...
2
votes
1answer
48 views

Simplify $\lim_{n\rightarrow \infty }\frac{((n+1)!)^{k+9}((k+9)(n))!}{((k+9)(n+1))!(n!)^{k+9}}|x|$

Where k is an unknown positive constant. I get to the point where $\lim_{n\rightarrow \infty }\frac{(n+1)^{k+9}((k+9)(n))!}{((k+9)(n+1))!}|x|$ but I'm not sure how I can further simplify or if this is ...
1
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1answer
53 views

Radius of convergence of a power serise involving the Fibonacci sequence.

Consider the power series $$\sum_{n=0}^{\infty}a_nz^n.$$ where, $a_0=0$ , $a_1=1$ , $a_n=a_{n-1}+a_{n-2}$. Find the radius of convergence of the power series. MY Attempt : Clearly $\{a_n\}$ is a ...
1
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0answers
49 views

Infinitely nested radicals

In a recent paper is is stated ( and maybe proved) that we can solve any polynomial equation with nested radicals. Here "nested radicals" means expression such as: $$ ...
3
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1answer
34 views

Power series converges to $\{0\}$ or $ \mathbb{R}$

Given this theorem: If a power series $\sum_{n=0}^\infty a_n x^n$ converges at some point $x_0 \in \mathbb{R}$, then it converges absolutely for any $x$ satisfying $|x| < \left|x_0\right|$. ...
1
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1answer
30 views

Conditions on coefficients of complex power series to ensure it is real

Given a complex valued function $f(z)=\sum_{n=0}^{\infty} a_nz^n$ with radius of convergence $R>0$, and $\rho\in (0,R)$, is there an if and only if giving that $f([0,\rho])\subset \mathbb{R}$? So ...
3
votes
2answers
56 views

Compute Power Series Convergence to a function

Consider the next power series $$ \sum_{n=1}^{\infty} \ln (n) z^n $$ Find the convergence radius and a the function $f$ to which the series converges. I have easily found that $R=1$ is the ...
0
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1answer
38 views

Does $\sum (2n)!/(n!) $ converge p-adically

Does $\sum (2n)!/(n!) $ converge p-adically, I have worked out $v_p((2n)!) \leqslant 2n/(p-1) $ similarly $v_p((n)!) \leqslant n/(p-1) $ I want to prove this using the result that it converges ...
1
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1answer
31 views

Radius of Convergence of Power Series $\sum_{n=0}^\infty\frac{\tanh^{(n)}(0)}{n!} z^n$

What is the radius of the power series $\sum_{n=0}^\infty\frac{\tanh^{(n)}(0)}{n!} z^n$? Justify your answer. My steps toward a solution I can express $\tanh$ simpler as: \begin{align*} \tanh z ...
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0answers
24 views

Frobenius Method

How would I solve it using the Frobenius method? I know how to solve problems with power series, but when considering the Frobenius method, I get stuck. Could someone please help me with this problem ...
4
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2answers
77 views

Convergence of $\sum \frac{(-1)^{n+1}}{n}z^n$ at $|z|=1$

I know that the power series $\sum \frac{(-1)^{n+1}}{n}z^n$ converges for $|z| \lt 1$ but I have been trying to determine what happens on $|z|=1$ Clearly the series converges at $z=1$ and diverges at ...
0
votes
2answers
43 views

Why is the radius convergence of $\sum_{n=0}^\infty \frac{x^{4n+1}}{4n+1}$ is $1$?

Why is the radius convergence of $\sum_{n=0}^\infty \frac{x^{4n+1}}{4n+1}$ is $1$? We know that $$\frac{1}{R} = \limsup_{n\to\infty} \sqrt[n] {\frac{1}{4n+1}} = 0$$ And therefore, $R=\infty$. ...
0
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2answers
39 views

To the power of n/2 - how to get rid of it.

In short, I have a formula $ \frac{2}{3}((-2)^{\frac{n}{2}}-1) $ I need to get rid of (n/2). I have to make sure that I only raise the number with degree n, the integer part, not n/2. Any ideas?
0
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3answers
30 views

Finding R for a power series

Let $\sum_2^\infty a_nx^n$ be a power series. Find the radius of convergence when $\lim \limits_{n \to \infty} \frac {a_n}{n^3}$ = 1. I've tried using root test but that gets messy, can't find a way ...
1
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1answer
43 views

Radius of convergence of a power series

Let $\sum_2^\infty a_nx^n$ be a power series. Find the radius of convergence when $\lim \limits_{n \to \infty} \frac {a_n}{3^n}$ = 1. There are a few more questions in this manner, but I'd like to ...
0
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2answers
58 views

Show divergence of this innocent series

Let $a$ be a complex constant with norm $|a| > 1$. Show that the series $$\sum_{k=1}^\infty \frac{a^k}{k^4}$$ diverges. The problem is really easy if $a \in \mathbb R$ is real: one just have to ...
1
vote
2answers
60 views

Find $\lim\limits_{n \to \infty} \frac{\log(1+2^n)}{\log(1+3^n)}$

How to calculate this limit? $$\lim\limits_{n \to \infty} \frac{\log(1+2^n)}{\log(1+3^n)}$$
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0answers
28 views

Identification of a series

Just a quick one, can anyone identify this power series in $x$, \begin{equation} \sum_{n \geq 0}\text{log}(\frac{r}{n})^{a}\frac{x^{n}}{n !} \end{equation} where $a,r$ are both constants? I have a ...
2
votes
2answers
27 views

Power series area of convergence with $\sin$

I want to examine for which $x\in\mathbb{R}$ the series $$\sum_{n=1}^{\infty}(2x-1)^n\sin\left(\frac{1}{n^2}\right)$$ converges. So far I have tried to use the inequality ...
0
votes
0answers
58 views

Radius of convergence $\sum_{n=1}^{\infty} \frac{x^n}{1+x^n} $

I want to prove that the radius of convergence of the series: $$\sum_{n=1}^{\infty} \frac{x^n}{1+x^n} $$ is $r=1$. Yeah in this interval does converge by comparison with the geo series ...
1
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3answers
89 views

$x+2$ is irreducible in the power series ring $\mathbb{Z}[[x]]$

For the last few days I am trying to prove that $x+2$ is irreducible in $\mathbb{Z}[[x]]$. I think that it is false... I would be very much thankful for any kind of suggestions and help.
1
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1answer
43 views

What function does this series represent?

A midterm I proctored recently showed that $$ \cos(\sqrt{x}) = \sum_{k=0}^{\infty} \dfrac{(-1)^k x^k}{(2k)!}$$ The question asked what function this series represents. It may represent cosine, but ...
0
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1answer
60 views

Power series divergence (real analysis)

Show that if a power series diverges at $x_0$ then it must also diverge when $\lvert x\rvert > \lvert x_0\rvert$ or provide a counterexample. I feel like there is a counterexample for some kind of ...
3
votes
1answer
19 views

problem convergent power series expansion such that $f^{(n)}(x)$ and $|f^{(n)}(x)|\leq 1$ for every $n\geq 1$ and for every $x\in(-1,1)$

Let $f:(-1,1)$ $\to \mathbb{R}$ such that $f^{(n)}(x)$ exists and $|f^{(n)}(x)|\leq 1$ for every $n\geq 1$ and for every $x\in(-1,1)$. Then f has a convergent power series expansion in a neighbourhood ...
1
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2answers
32 views

Non-trivial examples of power series which are uniformly convergent on $[0,1)$ and left-continuous at $x = 1$

The question is motivated by a more extensive problem that needs a formal proof, but I am not interested in help on the proof itself, but I'd like to see some examples of such power series. I put ...
4
votes
2answers
86 views

Prove $\lim\limits_{n \to \infty} \sup \left ( \frac{(2n - 1)^{2n - 1}}{2^{2n} (2n)!)} \right ) ^ {\frac 1 n} = \frac {e^2} 4$

This is a problem in Heuer (2009) "Lerbuch der Analysis Teil 1" on page 366. I assume that the proof should use $e = \sum\limits_{k = 0}^{\infty} \frac 1 {k!}$, but I cannot come further.
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1answer
41 views

finding the radius of convergence of a complex power series

I am trying to find radius of convergence of $$ \sum_{n=0}^{\infty} z^{a^n} $$ where $a>1$ integer. I obviously want to use $1/R = \limsup ( |c_n| )^{1/n}$. Is there a way to write $z^{a^n}$ ...
1
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0answers
17 views

Divergent Sum Renormalisation

I noticed an interesting property of holomorphic functions and I'm wondering if it forms the basis of divergent sum renormalisation. Let $f,g:\mathbb C \rightarrow \mathbb C$ be holomorphic ...
1
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1answer
80 views

Complex Arctan function and its power series

I face a sequence of confusing questions: In complex plane, note that $arctan(z)$ denote the principal branch of inverse complex tanget function ,by requiring $$\frac{-\pi}{2} < ...
3
votes
1answer
66 views

Is this correct reasoning about Taylor series?

Is the following correct reasoning about the Taylor series? I'm just trying to build some intuition but just want to make sure it's correct. If a function $f(x)$ has a power series representation ...
0
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2answers
53 views

Power Series (Laurent Series)

I need some help with this exercise: I need to obtain the power series development of this function: $$f(z)=\frac{\cos(z+1)}{(z^2-1)z}$$ Centered in $z_0=-1$ and valid in $z_1=\frac{1}{2}-i$ I know ...
1
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1answer
49 views

Help: infinite sum for matrices

Suppose $G$ is an $n\times n$ matrix. Can someone show me how you can carry $$I + G x^{-1} + G^2 x^{-2} + G^3 x^{-3} + G ^4 x^{-4} + \cdots$$ to $$(xI - G)^{-1}x$$ without having to "divide" ...
0
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2answers
43 views

Prove that $f$ has derivatives of all orders at $x=0$ [duplicate]

Let $\displaystyle f(x) = \begin{cases}e^{- \frac{1}{x^2}} &\text{for } x \neq 0 \\ 0 & \text{when } x=0 \end{cases}.$ Prove that $f$ has derivatives of all orders at $x=0$, and ...
0
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1answer
34 views

Maclaurin Series with Power in Denominator?

$$f(x) = \frac {x}{({125+6x^2})^{1/3}} $$ I'm having a bit of a tough time trying to figure out this question (in which I'm supposed to find the first five coefficients after creating a Maclaurin ...
1
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3answers
50 views

How to prove $\sum_{k=0}^{\infty}k^2x^{k} = \frac{x(1+x)}{(1-x)^3}\text{, }|x| < 1$? [duplicate]

How do I prove that the summation $$\sum_{k=0}^{\infty}k^2x^{k} = \dfrac{x(1+x)}{(1-x)^3}\text{, }|x| < 1\text{?}$$
2
votes
1answer
36 views

For every $z\in \Bbb C$, the exponetial series converges uniformly on every bounded subset of the complex plane

$$\operatorname{exp}(z)=\sum_{n=0}^\infty \frac{z^n}{n!}$$ This series converges uniformly on every bounded subset of the complex plane. What does this mean in simple terms?
1
vote
1answer
23 views

Exponential of a complex number converges absolutely

$$\operatorname{exp}(z)=\sum_{n=0}^\infty \frac{z^n}{n!}$$ This converges absolutely for every $z\in \Bbb C$. What does this mean to a layman?
4
votes
1answer
62 views

Series (Dilogarithm Function)

Let $\displaystyle f(x)=\sum_{n=1}^{\infty} \dfrac{x^n}{n^2} , \; x \in (0, 1)$. Evaluate $f(1/2)$ without using the known formulae of the dilogarithm or the equation it satisfies. May I have some ...
1
vote
1answer
39 views

The radius of convergence of a power series about a point interior to the domain of an analytic function

Let $f : \mathbb{R} \rightarrow \mathbb{R}$ be a real analytic function with domain an open, non-empty set $(a, b) \subseteq \mathbb{R}$, $-\infty \leq a < b \leq \infty$ and let $c \in (a, b)$. ...
0
votes
2answers
20 views

sequence power series simplification

Let $\{a_n\}$ be the sequence $a_n=\sqrt5\left(\frac{3+\sqrt5}2\right)^n - \sqrt5\left(\frac{3-\sqrt5}2\right)^n$ for each $n\ge 0$. Determine a rational expression for C(x) = $\sum_{n\ge0}a_nx^n$ ...