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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6
votes
2answers
146 views

How to show that $\lim\limits_{n\to\infty}n^{2/3}a_{n}=\sqrt[3]{2}/\Gamma{(1/3})$

Let $$\left(\dfrac{1+x}{1-x}\right)^{1/3}=\sum_{n=0}^{\infty}a_{n}x^n,|x|<1$$ Show that $$\lim_{n\to\infty}n^{2/3}a_{n}=\dfrac{\sqrt[3]{2}}{\Gamma{\left(\dfrac{1}{3}\right)}}$$
7
votes
1answer
118 views

Find the power series of $x\ln(1-x)$.

So the exercise I had to do was: Find the power representation of $x\ln(1-x)$. The way to go was finding the power series representation of $\ln(1-x)$ and then multiply it with $x$. But why can't you ...
0
votes
0answers
45 views

closed form expression for an infinite series

Is there any closed form expression for the infinite sum $∑_{n≥0}q^{n(n+1)/2}(1+q)(1+q^2)...(1+q^n)u^n$ where both q and n are variables and $n \in N∪0$?
0
votes
0answers
22 views

Sum of analytic functions

The sum of analytic functions is analytic. Does it mean that: $\sum_{i}\sum_{n=0}^{\infty}a_{in}x_{i}^{n} = \sum_{n=0}^{\infty}a_{n}x^{n}$ ? Is this also true $\sum_{i}\sum_{n=0}^{m}a_{in}x_{i}^{n} ...
0
votes
1answer
45 views

find radius of convergence

Suppose the radius of convergence of $\sum_n a_n x^n$ is $r$ ($r$ is a positive number). Prove that the radius of convergence of $\sum_n a_n^2 x^n$ is $r^2.$ I've tried to use Cauchy–Hadamard ...
3
votes
2answers
44 views

Show if the series $f(x)=\sum\limits_{k=1}^\infty \frac{1}{k} \sin(\frac{x}{k})$ converges uniformly or not.

So this is part of a different problem. The book and my professor say that the series $f(x)=\sum\limits_{k=1}^\infty \frac{1}{k} \sin(\frac{x}{k})$ converges uniformly on [0,1] by the Weierstrass ...
0
votes
0answers
14 views

Logarithm of the basic Lubin-Tate formal group

Let $K$ be a local field with finite residue field of cardinality $q$. Let $\pi$ be a uniformizer. The basic Lubin-Tate group (associated to $\pi$) is the unique formal group associated to the ...
8
votes
2answers
129 views

Convergence of differentiated power series

Consider $\displaystyle f(z)=\sum_{k=0}^\infty a_k z^k$ and suppose that $\displaystyle\sum_{n=0}^\infty f^{(n)}(0)$ converges. Prove that $\forall z \in \mathbb C, ...
0
votes
0answers
24 views

Lagrange Bürmann Inversion Series Example

I am trying to understand how one applies Lagrange Bürmann Inversion to solve an implicit equation in real variables(given that the equation satisfies the needed conditions). I have tried looking for ...
4
votes
0answers
81 views

Is there a known evaluation of $\sum_{k=1}^{\infty} \frac{1}{k^k}$? [duplicate]

Is there a known evaluation of $$\sum_{k=1}^{\infty} \frac{1}{k^k}$$ Wolfram Alpha says that it converges to $\approx 1.29129$.
0
votes
1answer
36 views

How to derive the formula to calculate the amount of cubes in a pyramid?

The pyramid looks like: For which I managed to derive the formula for the count of cube sides (ignoring the top). This was easy by simply thinking about it as a triangle: If we have 4 squares wide ...
3
votes
2answers
54 views

Convergence radius of power series is infinite

Which function is given by a power series whose convergence radius is infinite? $$A. \ \ \ e^{-\frac{1}{x^2}}$$ $$B. \ \ \ \sin{\left(\frac{1}{x}\right)}$$ $$C. \ \ \ ...
0
votes
0answers
32 views

Convergence of two power series

I just wanted to know, whether my results are correct. I should find the radius of convergency in both cases: $\sum_{n=1}^\infty \frac{z^{2n}}{n^23^{n}}$ with a quotient criterion ...
2
votes
0answers
48 views

How to power series expand determinants?

Say $g$ is a ($d\times d$) matrix which is given as, $g = g_0 + xg_2 + x^2 g_4 .. +x^{d/2 -1}g_{d-2}+ x^{d/2}(g_d + h_d(log (x)))$ where $d$ is an even number and each $g_i$ is a matrix (same ...
1
vote
2answers
110 views

Using series find $\int_0^1 \sqrt{1+x^4}\hspace{1mm} dx$ up to $2$ decimal places

I cannot figure out an aesthetic way to do this. Can someone give a beautiful solution to this ugly question? This is what I have tried yet. I used the fact that $$x = ...
0
votes
1answer
31 views

Radius of convergence of entire function

Let $f$ be an entire function on the complex plane. Is the radius of convergence of $f$ around any point $z_0$ infinite? If so, why? Thank you.
0
votes
1answer
18 views

Find the radius of convergence of $\sum_{n=0}^{\infty} (3^n + (-5)^n) x^{7n}$

I have to find the radius of convergence of the series $$\sum_{n=0}^{\infty} (3^n + (-5)^n) x^{7n}$$ I know that I will have something like $|x^7|<\frac{1}{L}$. I tried finding $R$ with ...
2
votes
0answers
40 views

Power series to solve differential equations?

We can use the formula $$F(x)=e^{λx} [ ρ-λμ-\dfrac{1}{2} λ^2 σ^2 ]^{-1}. (1) $$ to derive an expression for F(x) when f(x) is any integer power $x^n$. Begin by observing that for the ...
2
votes
1answer
45 views

Showing integral on contour tends to zero

I'm trying to prove: $$\int \frac{e^{t(z+\frac{1}{z})}}{z^2} = \sum_0 ^{\infty} b_m t^{2m+1}$$ Where the integral is over a contour centre the origin, radius R, and the $b_m$ are some coefficients. ...
8
votes
4answers
210 views

Real-analytic $f(z)=f(\sqrt z) + f(-\sqrt z)$?

Are there nonconstant real-analytic functions $f(z)$ such that $$ f(z)=f(\sqrt z) + f(-\sqrt z)$$ is satisfied near the real line ? Also can such functions be entire ? And/Or can they be periodic ...
0
votes
0answers
19 views

restriction on the coefficients $a_n$ if entire function $\sum_{n=0}^\infty a_n x^n << e^x$

I need to find restrictions on coefficients $a_n$ such that the entire function $$f(x)=\sum_{n=0}^\infty a_n x^n $$ Satisfies $$f(x) \ll e^x \text{ }(x \to \infty)$$ Since ...
0
votes
1answer
37 views

Taylor Series expansion and first four terms of $7x^2 e^{-4x}$

As the series I got $$ \sum_{n=0}^\infty (-1)^n(4x)^n/n! $$ which I think is right. However, I am not sure how to get the first four non zero terms.
1
vote
1answer
16 views

Power Series Simplification

$N$ is Poisson with parameter $z$. Find $E[N\cdot(N-1)\cdot(N-2)\cdots(N-k+1)]$ The answer is apparently $z^k$. $$E[N\cdot(N-1)\cdot(N-2)\cdots(N-k+1)]=E\left[\frac{N!}{k!}\right]=\sum_{n\geq0} ...
1
vote
0answers
36 views

Generalizations of $\sum_{m=3n+2}^{\infty}\phi^m=\phi^{3n}$ and $\sum_{m=13n+1}^{\infty}(\sqrt2-1)^m=\dfrac{(\sqrt2-1)^{13n}}{\sqrt2}$

I noticed that the following identies hold with the help of wolfram alpha and oeis. I'm sure they're well-known, but I'd like to know how they generalize. ...
1
vote
2answers
30 views

Meaning of interval of convergence when approximating functions

Let's say I have a Taylor series approximation, $p(x)$, of a function $f(x)$ at $a$: $$ p(x)=\sum_{n=0}^\infty{\frac{f^{(n)}(a)}{n!}(x-a)^n} $$ And that this Taylor series has a radius of ...
0
votes
1answer
34 views

Sum of a function involving $n-$root

I'm trying to find the series of $$f(x)=\sqrt{1-x^{3}}$$ Can I just use the fact that $$\frac{1}{1-x}=\sum x^{n},\quad|x|<1$$ writting $x^{3}$ in the place of $x$ and then, getting this: ...
2
votes
1answer
39 views

Exponential function as a sum

I have an exercise that asks me to write $e^{2x}$ using a power series of $x+1$. I know that $$e^{2x}=\sum_{n=0}^{\infty}\frac{(2x)^{n}}{n!}$$ Then, I tried something like this $$x=y+1\Rightarrow ...
0
votes
1answer
51 views

Radius and interval of convergence of the power series $\sum 2^{n^2}x^{n!}$?

How to calculate the radius and interval of convergence of the following series: $$\sum 2^{n^2}x^{n!}$$ The formula for the radius is: $$R = \frac{1}{\limsup_{n\to\infty} \sqrt[n]{|a_n|}}$$ or ...
1
vote
0answers
46 views

Sum of a power series

I have to find the sum of this series $$\sum_{n=1}^{+\infty}\frac{x^{n-1}}{n!}$$ Using integral, I got $$\int\sum_{n=1}^{+\infty}\frac{x^{n-1}}{n!}=\sum_{n=1}^{+\infty}\frac{x^{n}}{n\cdot n!}$$ I ...
5
votes
4answers
96 views

Power series convergence radius

My question is: how do I calculate the radius convergence of a power series when the series is not written like $$\sum a_{n}x^{n}?$$ I have this series: $$\sum\frac{x^{2n+1}}{(-3)^{n}}$$ Can I use the ...
1
vote
1answer
19 views

Convergence of series, using big oh or little oh notation.

Let $p\in \mathbb{R}$ and $a_n=(e-(1+1/n)^n)^p$. For which $p$ will $\sum_{n=1}^{\infty} a_n$ converges? Because of the "additive look" of $a_n$, I tried to use taylor expansion and big oh, little oh ...
3
votes
2answers
204 views

Why is the domain of convergence of a power series a perfect disk?

I've been going over power series in my Differential Equations class for approximating solutions, and one thing that's been fascinating me is the statement that there is a radius of convergence, ...
0
votes
3answers
77 views

Represent $\frac1{(1+x)^2}$ as a power series

Represent $\frac1{(1+x)^2}$ as a power series. if we put $\frac1{1+2x+x^2}$: $$ \sum _{n=1}^{+\infty}\left(-x-x^2\right)^n $$ and if we differentiate $\frac{-1}{1+x}$: $$ -\sum ...
1
vote
1answer
72 views

Some exam question on power series convergence

I provide my solution to the problem and wonder if I was thinking in a correct way. Find the radius of convergence of $$\sum_{n=0}{1 \over 1+n3^n}z^n$$ and give with reasoning a point $z_0$ on the ...
1
vote
3answers
57 views

calculate radius of convergence

Let $\{a_n\}_{n=0}^{\infty}$ be sequence such that $$a_1 = a_0 = 1$$ $$a_{n+1}=a_n+ a_{n-1}$$ show that the radius of convergence of $\sum\limits_{n=0}^{\infty \:}a_nx^n$ is ...
1
vote
0answers
61 views

How to generilize the the following summation.

While searching for a summation formula I come accross the following equation on wikipedia Equation $$\sum\limits_{k=1}^{n}{k^m z^k}=\left(z\frac{d}{dz}\right)^m\frac{z-z^{n+1}}{1-z}$$ So I tried to ...
0
votes
1answer
26 views

A basic question about the radius of convergence of infinite power series.

I have a somewhat theoretical question to the definition of the radius radius of convergence of infinite power series. According to the definition for a power series $\sum_{n=0}^\infty a_nx^n$ radius ...
1
vote
3answers
61 views

Laurent series and residue of $f(x)=\frac{1}{1+e^z}$

I am having trouble trying to expand this function using Laurent series, and finding the residue$$f(x)=\frac{1}{1+e^z}$$ If I replace $e^z$ with its series I get ...
10
votes
6answers
3k views

Why infinity multiplied by zero was considered zero here?!

I watched an online video lecture by some professor and she was solving a convergence problem of the power series $$\sum_{n=1}^\infty n!x^n,$$ i.e., she was finding the values of $x$ for which this ...
0
votes
1answer
14 views

Why not shift the index of the derivative in Euler series?

I'm reading over solving linear differential equations with analytic coefficients, and finding the solutions that are near regular singular points. In the earlier section on solving similar equations ...
1
vote
2answers
44 views

Calculate the radius of convergence [closed]

Being $\sum _{n=0}^{\infty \:}a_n\cdot x^n$, $a_1=a_0=1$ $a_{n+1}=a_n + a_{n-1}$ show that the radius of convergence is $\dfrac{-1+\sqrt{5}}{2}$ Thanks!
3
votes
2answers
121 views

Calculate the value of $\sum\limits _{n=1}^{\infty }\:\dfrac{n}{2^n}$ [closed]

In a previous question it is asked to represent $f(x)=\dfrac{x}{1-x^2}$ as a power series. It gave me $\displaystyle\sum _{n=1}^{\infty \:}x\left(2x^2-x^4\right)^{n-1}$. Then they ask to use the last ...
1
vote
0answers
28 views

Rational approximation or series expansion of $K_0$ and $K_1$ for small z

I'm looking for a series expansion of the modified Bessel functions of second kind $K_0(z)$ and $K_1(z)$ for $$|z|<5, ~~|phase(z)| < \pi$$ My $z$ can be described as $z = a\cdot \sqrt{ix}$, ...
3
votes
3answers
65 views

how to represent $\int \frac{\arctan \left(x\right)}{x}dx$ as a power series?

I have no idea. I don t even no how to calculate the primitive can you help me?
3
votes
2answers
44 views

How to represent $\ln(5-x)$ as a power series?

I know that $$ \ln(1+x)=\sum _{n=1}^{\infty }\:\left(-1\right)^{n-1}\frac{x^{n}}{n} $$
0
votes
2answers
27 views

finding the residue of the following

I can find $$Res_{z=0} \frac{\sin z}{z^4} $$ but stuck with finding $$Res_{z=0} \frac{\cot z}{z^4} $$ so please help me
0
votes
2answers
39 views

By completing the square, show that $\int_{0}^{\frac{1}{2}}\frac{dx}{x^2-x+1}=\frac{\pi }{3\sqrt{3}}$

By completing the square, show that $\int_{0}^{\frac{1}{2}}\frac{dx}{x^2-x+1}=\frac{\pi }{3\sqrt{3}}$. I found that $\int_{0}^{\frac{1}{2}}\frac{dx}{x^2-x+1}$ equals to ...
0
votes
1answer
41 views

The power series $\sum_0^\infty2^{-n}z^{2n}$ converges, if

The power series $\sum_0^\infty2^{-n}z^{2n}$ converges, if a) |$z$| $\le$ 2. b) |$z$| $<$ 2. c) |$z$| $\le$ $\sqrt{2}$. d) |$z$| $<$ $\sqrt{2}$. Please anyone give me the answer. I think ...
3
votes
2answers
35 views

Help with understanding a proof about differentiating a real power series

I'm stuck trying to understand a proof of the following theorem: Let $ \sum a_nx^n$ be a power series with radius of convergence $ R $. Then $ \sum na_nx^{n-1}$ also has radius of convergence $ R $. ...
0
votes
3answers
77 views

If $f(x) = e^{x^{2}}$, show that $f^{(2n)}(0)=(2n)!/n!$ [closed]

If $f(x) = e^{x^{2}}$, show that $f^{(2n)}(0)=(2n)!/n!$