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
130 views

Find the radius of convergence for $\sum^{\infty}_0 n^nz^{n^n}$

Find the radius of convergence for $\sum^{\infty}_{n=0} n^nz^{n^n}$ This is not a power series, but if I define $a_k=k$ if $k=n^n$ and $a_k=0$ otherwise, I would have a power series such that ...
2
votes
1answer
363 views

Reversion of power series

So, I just heard about this method. How does one determine the coefficients, and what is it used for? For example, given $$ y = x - \frac{x^3}{6} + \frac{x^5}{120} + O(x^7)$$ reversion would give a ...
1
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2answers
118 views

Is there a real power series with radius of convergence 1 that converges at 1 but not at -1?

I can find a power series that has radius of convergence 1 but since any series that converges absolutely converges, I cannot find any that converges at 1 but diverges at -1... Can you help me? Thank ...
3
votes
1answer
255 views

Convergence radius of $\sqrt{\cos(z)}$

Compute the first 3 non zero terms of the Taylor expansion of $\sqrt{\cos(z)}$ at $z=0$ and determine its convergence radius, considering only the principal branch of the square root. I've computed ...
3
votes
1answer
58 views

Analytic function satisfying $x^2f''(x) +xf'(x)+x^2f(x)=0$ and $f(0)=1$

Suppose that the power series $\sum_{n=0}^\infty a_nx^n$ converges for all real $x$ to a function $f(x)$ that satisfies $$x^2f''(x) +xf'(x)+x^2f(x)=0 \quad \text{and} \quad f(0)=1.$$ ...
2
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2answers
89 views

Formal power series with all derivatives zero

I have the following question. Suppose I have a formal power series $f(x)=\sum\limits_{i=0}^\infty c_ix^i$ with real coefficients. Suppose that all the derivatives ...
1
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1answer
112 views

Convergence radius: $R = \lim_{n \rightarrow \infty} \frac {\mid a_n \mid} {\mid a_{n+1} \mid}$ (incl. $\infty$) when $R = 0$ and Ratio test

I have read the following proof of a theorem in a textbook of mine, and I've been wondering why the proof holds when $$R = \lim_{n \rightarrow \infty} \frac {\mid a_n \mid} {\mid a_{n+1} \mid} = 0$$ ...
24
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1answer
823 views

Is it possible for a function to be smooth everywhere, analytic nowhere, yet Taylor series at any point converges in a nonzero radius?

It is well-known that the function $$f(x) = \begin{cases} e^{-1/x^2}, \mbox{if } x \ne 0 \\ 0, \mbox{if } x = 0\end{cases}$$ is smooth everywhere, yet not analytic at $x = 0$. In particular, its ...
6
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1answer
111 views

Simplify $\sum_{n=0}^{N}\binom{N}{n} \frac{a^{N-n}}{n!} \frac{d^n}{dx^n} f(x)$

Simplify the following expression $$S_N = \sum_{n=0}^{N}\binom{N}{n} \frac{a^{N-n}}{n!} \frac{d^n}{dx^n} f(x), $$ where $a$ is a real number and $f(x)$ is an analytic real function. What is $\lim_n ...
0
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2answers
149 views

Maclaurin Series confusion

Using the Maclaurin expansion formula: to find the Maclaurin series for $sin(3x)$, I can get the correct answer by using $x^n$ in the formula above (in the tail-end of the formula). Similarly, to ...
4
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2answers
140 views

How to find the radius of convergence?

The function is $\dfrac {z-z^3}{\sin {\pi z}} $. How to find the radius of convergence in $ z=0 $?
3
votes
1answer
117 views

Finding complex power series with interesting boundary behavior

I need to find one (or more) interesting complex power series to give to my students for their analysis exam. Ideally, this would be a power series that has interesting behavior at the boundary, i.e. ...
12
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2answers
355 views

Finding the derivative of $x\uparrow\uparrow n$

I am trying to find a general derivative for the function: $f(x)=x^{x^{x^{...^{x}}}}$however to do that I must find $f^{\prime }$ and $f^{\prime \prime}$...etc. I am now trying to write down a general ...
1
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2answers
58 views

An easy question on complex

Let $\{u_{k}\}_{k=1}^{\infty}$ be a complex number sequence. If $\sum_{k=1}^{\infty}\lambda^{k}u_{k}=0$, for each $\lambda\in \mathbb{D}(0, 1/3)$(where the $\mathbb{D}(0, 1/3)~$denotes an open disc ...
1
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1answer
104 views

Prove $\sum_{m \geq 1} {\frac{(2m-2)!}{(1-\rho)\cdots(m-\rho)} \frac{t^m}{(1-x)^{2m-1}}} $is divergent

How do I show that the following power series is divergent? $$ u(t,x) = \sum_{m \geq 1} {\dfrac{(2m-2)!}{(1-\rho)\cdots(m-\rho)} \dfrac{t^m}{(1-x)^{2m-1}}} $$ where $t$ is complex 1-dimensional, $x$ ...
4
votes
2answers
424 views

Maclaurin expansion of arctan: convergence?

In my textbook, the Maclaurin series expansion of $\arctan{x}$ is found by integrating a geometric series, that is, by noting that $\frac{d}{dx}(\arctan(x)) = \frac{1}{x^2+1}$ then rewriting the ...
4
votes
2answers
133 views

meromorphic function in the unit disc with only one pole of order n

Let $f$ be meromorphic in a neighborhood of $\{|z| \leq 1\}\setminus \{1/2\}$ and have a pole or order $n$ at $1/2$. Suppose that $|f| < 3$ on $\{|z|=1\}$. Show that for any $\phi \in \mathbb{R}$, ...
1
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2answers
71 views

Limit of the general term of a power series in a pole

Let $\Omega = D(0,2)/\{\frac{1}{2}\}$ , where $D(0,2)$ is a disc, $f$ holomorphic in $\Omega$. $\frac{1}{2}$ is a simple pole for $f$ with residue $1$, calculate $$ lim_{n \to \infty} ...
2
votes
3answers
99 views

Radius and domain of convergence for $\sum _{n=1}^{\infty}2^n x^{n^2}$

Another Question on Radius of convergence : Calculate Radius and domain of convergence for $$\sum _{n=1}^{\infty}2^n x^{n^2}$$ I used the formula $\lim_{n\rightarrow \infty} |\frac{a_n}{a_{n+1}}|$ ...
3
votes
2answers
104 views

Radius of convergence and domain of convergence

Question is to calculate Radius and domain of convergence for : $$\sum_{n=1}^{\infty}(\arctan\frac{1}{x})^{n^2}$$ What I have tried is : Radius of convergence is $1$ I am sure about this. Coming ...
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2answers
50 views

a differential equation equation related to fourier series

I am really struggling with this one. Any help is welcome! For equation $f''(z) + p(z) f'(z) + q(z) f(z) = 0$, where $p(z)$ and $q(z)$ are fixed polynomials. Given $f(0)=f_0$, $f'(0)=f_1$, prove that ...
2
votes
2answers
123 views

Unconventional way, how to expand to Maclaurin series

Let's have function $f$ defined by: $$f(x)=2\sum_{k=1}^{\infty}\frac{e^{kx}}{k^3}-x\sum_{k=1}^{\infty}\frac{e^{kx}}{k^2},\quad x\in(-2\pi,0\,\rangle$$ My question: Can somebody expand it into a ...
1
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1answer
50 views

Exact expression for series

Can the exact expression for the following series be found, given $|x|<1$? Just curious. $f(x) = \frac{x^2}{17}+\frac{x^3}{3}+\frac{x^4}{3}+\frac{x^5}{3}+ \ldots$
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2answers
84 views

Series Solution of Second Order Linear Equation, IVP

Consider the initial value problem $$y' = \sqrt{1-y^2}$$ $$y(0) = 0$$ Look for a solution of the IVP in the form of power series about x=0. I have started with assuming that $ y = \sum_{n = ...
1
vote
1answer
89 views

Maclaurin Series of $\int_0^x \cos t^2\,dt$

Find the Maclaurin Series for $\int_{0}^{x}\cos t^2\,dt$. $$\cos(x) = \sum\frac{(-1)^n x^{2n}}{2n!}$$ I'm trying this: $$\cos^2 x = \sum\frac{(-1)^n x^{4n}}{(2n!)^2}$$ How would you solve this ...
1
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3answers
534 views

First $3$ non-zero terms of the Maclaurin Series $\frac{1}{\sqrt{4+x^3}}$

Since each derivative will be multiplied by $3x^2$, are all the terms of this Maclaurin series $0$?
1
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1answer
66 views

Expansion of Jacobi's $\theta_3(0,q)$ in q=1

In trying to solve a certain limit, I wondered how Mathematica comes up with this weird expression for a series expansion of Jacobi's $\theta_3(0,q)$ in $q=1$ at the order 0: $\frac{i \sqrt{\pi } ...
0
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1answer
101 views

Finding the interval of convergence of $\sum_{n=1}^\infty ((-1)^n2^n(x-3)^nn!)/(n^n)$

$$\sum_{n=1}^\infty (-1)^n \frac{n!(2^n)(x-3)^n}{n^n} $$ this is what I tried: $$ \lim_{n-> \infty} \left| \frac{(n+1)n!2^{(n+1)}(x-3)^{(n+1)}}{(n+1)^{(n+1)}} \cdot \ \frac{n^n}{n!2^n(x-3)^n} ...
1
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1answer
60 views

Power series: If $|z-z_0|< R$ the series converges absolutely

I'm trying to prove absolute convergence of the power seris $$\sum_{n=0}^{\infty} a_n (z-z_0)^n, \qquad |z-z_0| < R$$ where $R^{-1} = \limsup |a_n|^{1/n}$. WLOG, suppose $z_0=0$ (otherwise ...
2
votes
1answer
70 views

Show that $\sum_{n=0}^\infty a_n z^n$ converges $\forall z\in\mathbb{C}.$

Assume that $\sum_{n=0}^\infty b_n z^n$ converges $\forall z\in\mathbb{C}.$ Let $x=\lim_{ n\rightarrow\infty}|\frac{a_n}{b_n}|$ exists. Show that $\sum_{n=0}^\infty a_n z^n$ converges $\forall ...
9
votes
1answer
512 views

$f'/f\in\mathbb{Z}[[x]]$ for polynomials vs. formal power series $f$

I am curious about the following problem from MIT's Problem Solving Seminar (#26 here, though the link may stop working after a few weeks): Let $f(x) = a_0+a_1x+\cdots\in\mathbb{Z}[[x]]$ be a ...
1
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1answer
41 views

Not quite alternating series

Quite a lot of things are known about alternating power series $$ \sum_{n \geq 0} (-1)^n a_n x^n, \quad a_n > 0, $$ like closed-form expressions for well-chosen $a_n$ and so on. In a problem I'm ...
1
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1answer
27 views

Equality between a fonction and an infinite serie.

I want to prove this equality : -ln(x+1) = S(from 1 to infinite) ((-x)^n/n) //for x between -1 and 1 (not included of course). I wrote on my notebook : show the derivates are equals and the two ...
1
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3answers
124 views

Radius of Convergence - $\sum_{n=1}^{\infty}2^n x^{n^2}$

What is the radius of convergence of this power series here? $$\sum_{n=1}^{\infty}2^n x^{n^2}$$
1
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1answer
1k views

Express the indefinite integral $\int\sin x^2~dx$ as a power series

What does this mean? I never saw this in my class/notes so I don't understand the conversion from integral to power series. Also if the integral were defined from $0$ to $1$, what new steps do I add? ...
0
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0answers
44 views

Is something wrong in this proof?

Show that if $\sum a_nx^n$ has convergence radio $R$ and $\limsup |a_n| > 0 $, then $R\leq 1$. Proof: Suppose that $\sum a_nx^n$ has convergence radio $R$ and $\limsup |a_n|=\alpha > 0$. ...
0
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1answer
61 views

Real Analysis: Proof of Radius of Convergence

I just would like to know are my steps right. $\textbf{I needed to show that the radius of convergence of $\sum_{n=0}^{\infty}a_nz^n$ is the same a that of $\sum_{n=0}^{\infty}na_nz^n$.}$
1
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1answer
61 views

Question about convergence of a power series and when the series is not zero

Following is a past exam question I am trying to solve as a preparation for my own exam. Let $(a_n)_{n\in\mathbb{N}}$ be a sequence of real numbers with $a_n \leq M$ for some $M\in\mathbb{R}^{+}$ and ...
1
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3answers
69 views

$S(t) = \sum_{n=1}^{+\infty}\frac{e^{int}}{n}$

Let $t \in (0, 2\pi)$, calculate $S(t) = \sum_{n=1}^{+\infty}\frac{e^{int}}{n}$. If we can derive term by term is easy, but how to prove that it is possible ?
3
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1answer
42 views

$\sum_{n=0}^{+\infty}|a_{n}-a_{n+1}| $ converges $\Rightarrow \sum_{n=0}^{+\infty}a_{n}z^{n}$ converges

Let $\{a\}_{n} \subset \mathbb{C}$ , $a_{n} \rightarrow 0 $ , $\sum_{n=0}^{+\infty}|a_{n}-a_{n+1}| < +\infty$; then show that $\sum_{n=0}^{+\infty}a_{n}z^{n}$ converges if $|z|\le 1 \ $ , $z \ne ...
2
votes
2answers
59 views

Power series of trigonometric functions

Problem statement: Determine those $x$, for which the power series is convergent and determine the sum. $$f(x)=x+\sum_{n=2}^{\infty}(-1)^{n-1}2n\frac{x^{2n-1}}{(2n-1)!}$$. Progress: I have ...
0
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3answers
71 views

Prove the Identity For Fringe Patterns

Prove the two Identities for $-1 < r < 1$ $$\sum_{n=0}^{\infty} r^n\cos n\theta =\frac{1-r\cos\theta}{1-2r\cos\theta+r^2}$$ $$\sum_{n=0}^{\infty} r^n\sin{n\theta}=\frac{r \sin\theta ...
2
votes
3answers
54 views

A power series from $\frac{x}{9+x^2}$

I need to make power series from $\frac{x}{9+x^2}$, and I don't have any idea how. The only thing I know is how to make power series from $\frac{1}{1-q}$. Thank you!
1
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2answers
30 views

Help with power series $f(x)=\frac{2x}{(1-x^2)^2}=\frac{d}{dx}\left(\frac{1}{1-x^2}\right)$

Given that $f(x)=\frac{2x}{(1-x^2)^2}=\frac{d}{dx}\left(\frac{1}{1-x^2}\right)$, find a power series for $f(x)$. What is its radius of convergence? So far I got the following: ...
1
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2answers
190 views

Uniform convergence of the series for $\frac{x}{\sqrt{1+x}} = \sum_{n=1}^\infty (-1)^n\binom{-1/2}{n}\left(\frac{x}{1+x}\right)^{n+1}$

I am looking for the values where this series expansion converges uniformly. $$\frac{x}{\sqrt{1+x}} = \sum_{n=1}^\infty (-1)^n\binom{-1/2}{n}\left(\frac{x}{1+x}\right)^{n+1}$$ Intuitively, I believe ...
5
votes
3answers
389 views

Radius of convergence and the endpoints of a power series

Find the radius of convergence and the convergence at the end points of the series: $$\sum_{n=1}^\infty(2+(-1)^n)^nx^n$$ This is what I did: $a_n=(2+(-1)^n)^n\Rightarrow ...
5
votes
1answer
103 views

Find a power series for the following function $\frac{x^2}{1+x^3}$

I'm not sure if I solved this correctly, but here is the problem: Find a power series for the following function $\frac{x^2}{1+x^3}$ And here is what I did: $$x^2\frac{1}{1-(-x^3)}$$ Here is where ...
4
votes
3answers
264 views

Asymptotic expansion of $J(t) = \int^{\infty}_{0}{\exp(-t(x + 4/(x+1)))}\, dx$

I want to derive an asymptotic expansion for the following Bessel function. I think I need to rewrite it in another form, from which I can integrate it by parts. I am interested in obtaining the ...
1
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0answers
1k views

Taylor's Formula vs. Taylor's Inequality

In my calculus book, Essential Calculus, and in class we were using Taylor's formula to approximate the remainder in Taylor polynomials but I am having a bit of trouble understanding the intuition ...
0
votes
1answer
80 views

radius of convergence of the power series $\sum_{n=0}^{\infty} z^{n!}$

How to find the radius of convergence of the power series $$\sum_{n=0}^{\infty} z^{n!}?$$ I don't know how to start !!!