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).

learn more… | top users | synonyms

2
votes
3answers
37 views

Find which differentiable function is determined by this power series

Given the series $$\sum\limits_{n=1}^\infty \dfrac{1}{n(n+1)} (\sin(x))^n$$ for all $x \in \mathbb{R}$, find an interval on which it determines a differentiable function of $x$, together with an ...
1
vote
2answers
53 views

writing $\ln(1+x)$ as power series

\begin{align*} \left[\ln\left(1+x\right)\right]' &= \frac{1}{1+x}\\ &= \frac{1}{1-(-x)}\\ ...
1
vote
0answers
28 views

Power series expansion of $e^{\frac{c}{2}(z-1/z)}$

Show that \begin{equation} e^{\frac{c}{2}(z-1/z)}=\sum_{n=-\infty}^{\infty}a_nz^n \end{equation} where \begin{equation} a_n:=\frac{1}{2\pi}\int_0^{2\pi}\cos(n\theta-c\sin(\theta))d\theta ...
0
votes
3answers
40 views

Skip terms in power series of $\cosh$ and $\sinh$

Is there a way to skip every second term in the power series representation of $\sinh{x}$ and $\cosh{x}$ and adjust the other terms accordingly (approx.)? So, instead of $$\sinh{x} \approx x + ...
0
votes
2answers
29 views

Power Series and their radii of convergence

Suppose that $$\sum\limits_{n=0}^\infty a_nx^n \ \ and \ \ \sum\limits_{n=0}^\infty b_nx^n$$ $R$ and $S$ respectively.Let $U$ be the radius of convergence of $$\sum\limits_{n=0}^\infty c_nx^n$$ ...
1
vote
2answers
49 views

Use power series $\sum^{\infty}_{n=1} \frac {z^{n+1}} {n(n+1)}$ to show $\sum^{\infty}_{n=1} \frac {1} {n(n+1)} =1$.

Consider $\sum^{\infty}_{n=1} \frac {z^{n+1}} {n(n+1)}$ (power series). I've found that the sum-function $g(z) := \sum^{\infty}_{n=1} \frac {z^{n+1}} {n(n+1)}$ is defined and continuous on $|z| \le ...
0
votes
1answer
18 views

Let $R$ be radius of convergence for $\sum^{\infty}_{n=0} a_n z^n$. find radius of convergence for $\sum^{\infty}_{n=0} a_n z^{kn+l}$ in terms of $R$.

Let $R \in [0,\infty)$ be radius of convergence for $$\sum^{\infty}_{n=0} a_n z^n$$. For $k \in \mathbb N, l \in \mathbb N_0$ find radius of convergence for $$\sum^{\infty}_{n=0} a_n z^{kn+l}$$ in ...
1
vote
1answer
22 views

Let $\sum^{\infty}_{n=1} \frac {z^{n+1}} {n(n+1)}$ be a power series. Show sum-function $g(z)$ is continuous on $|z|\le 1$.

Let $\sum^{\infty}_{n=1} \frac {z^{n+1}} {n(n+1)}$ be a power series. I've shown that radius of convergence is $R=1$. I've a theorem saying that the sum-function $g(z)=\sum^{\infty}_{n=1} \frac ...
1
vote
1answer
26 views

Determine the maximal compact interval such that $\sum_{n=0}^{\infty}(-1)^n\frac{x^{2n+1}}{2n+1} = \arctan(x)$ holds true

The Assignment: Determine the maximal compact interval, such that the following identity holds true:$$\sum_{n=0}^{\infty}(-1)^n\frac{x^{2n+1}}{2n+1} = \arctan(x)$$ Explain your answer and show ...
0
votes
2answers
30 views

Why does csc(z) only have 1st order poles?

$\csc{z} = \frac{1}{\sin{z}}$ is said (in my text book) to have only simple (1st order) poles. I can see that this is justified since the Laurent series expansion is: $$ \csc{z} = \frac{1}{z} + ...
0
votes
2answers
57 views

Prove $\sum_{n = 1}^{\infty} 2^{-n} x^n$ does not converge uniformly on $(-2, 2)$

How can one go about proving this? (I understand that the said series does converge uniformly on all $[-a, a]$ where $0 \leq a < 2$.) I am especially interested in knowing if there is a way to ...
1
vote
3answers
34 views

The series of $\sum\limits_{j=0}^{\infty} (\frac{1}{2})^{2j}$

The series converge: $$\sum_{j=0}^{\infty} \left( \frac{1}{2} \right)^{2j}$$ I try to put it in geometric series but I am stuck some help please.
0
votes
1answer
20 views

Laurent series of quotient

If I have two functions $f,g$ that are holomorphic around a point $z_0 \in \mathbb{C}$. Assume the Laurent series are known and both $f$ and $g$ have a finite principal part. $$f(z) = ...
2
votes
0answers
33 views

Find maximal possible sum of a tricky series

for $a \in R$, $n \in N$ let $a_n$ closest distance between $a$ and $\frac m {2^n}$, where $m \in Z$. Find maximal possible sum of a series: $\sum_{n=0}^\infty a_n$ I came up with solution for the ...
2
votes
1answer
52 views

If $n^3 < |a_n| < n^4$ find the radius of convergence for $\sum_{n=2}^\infty a_nx^n$

If $n^3 < |a_n| < n^4$ find the radius of convergence for $\sum_{n=2}^\infty a_nx^n$ Could someone explain how he got inequality (1)? Theorem 4.1 stated that a power series converges if $|x| ...
0
votes
1answer
60 views

Analysis of singularities and taylor representation of $f(z)=\frac{z^2-1}{\sin \pi z}$

Let $$f(z)=\frac{z^2-1}{\sin \pi z}$$ A) Find all singulartities of $f$ in $\mathbb{C}$ and classify each as a pole (specifying the order), essential, removable, or other. B) Explain why $f(z)$ has ...
2
votes
1answer
78 views

Simplification trick

it is maybe at bit of a silly question, but one of our professors wrote the following equations and I would like to know what exactly he did. I'm sure it is something easy but I have no clue: ...
1
vote
1answer
37 views

Calculate the value of the integral of a series

let $$P(r,\varphi):= \dfrac{1}{2\pi} \sum_{n \in \mathbb{Z}} r^{|n|}e^{in\varphi} $$ with $\varphi \in \mathbb{R}$ and $ 0< r <1$. Prove that $$\int_{0}^{2\pi}P(r,\varphi)d\varphi =1$$ My ...
1
vote
1answer
23 views

For what interval does this power series converge and for what interval does it determine a differentiable function?

For what range of values of $x$ does $\sum_{n=1}^{\infty } \dfrac{1}{n}(1+\sin x)^n$ converge? Find with proof an interval on which it determines a differentiable function of $x$ and show that ...
0
votes
1answer
29 views

Radius of Convergence ratio test

using the ratio test for the following sum from n = 0 to infinity of $$ \sum_{m=0}^{+\infty}\frac{(-1)^m}{(m!)^2} x^{2m +10} $$ I need to find the radius of convergence. I managed to get up to ...
1
vote
0answers
24 views

Somehow “mirroring” the Taylor-expansion of some $g(x)$

In proceeding with an older discussion of a summation-procedure for divergent series using the matrix of Eulerian numbers I came to a rather general formulation but cannot/do-not-know-how-to make ...
3
votes
0answers
54 views

Prove that $e^{\ln{z}}=z$ from the power series

For my course in complex analysis we have to prove that the trivial relation $e^{\ln{z}}=z$. We are given the series for $\ln z$: $$f(w)=\sum_{n=0}^\infty (-1)^{n+1}\frac{w^n}{n}$$ $$\ln z = ...
0
votes
0answers
40 views

Properties of power series and their analytic continuation

Suppose a power series $$\sum_{k=0}^\infty a_k z^k$$ is valid for $|z|<R$, and can be analytically continued to some function $f(z)$, for all $z\in\mathbb{C}$ , except for a finite number of points ...
1
vote
0answers
24 views

What is Radius of Convergence used for?

What is the applications for "Radius of convergence"? I haven't been successful in finding any information about the applications, just a lot of information about how to calculate and what it is... ...
4
votes
1answer
42 views

Power series with differentiable coefficients

Suppose for each $s$ in an open interval, $P_s(x)=\sum_{k=0}^\infty a_k(s) x^k$ is a power series with radius of convergence greater than R, where each $a_k(s)$ is differentiable. My question is: Is ...
2
votes
1answer
24 views

Infinite differentiability and power series expansion

Does every infinitely differentiable function have a power series expansion?Is this a theorem? Or is this an open question?
3
votes
0answers
50 views

How come Stone-Weierstrass theorem does not imply that in a given interval every continuous function has a power series expansion?

Since for all continuous functions we get a polynomial sequence that uniformly converges to that function? As the degree of polynomial increases it should look like a power series expansion?
1
vote
1answer
52 views

$\frac{1}{(1+s^{2}) (1+t^{2})}$ real analytic in $\mathbb R^{2}$ but not real-entire; why?

A complex valued function $F,$ defined on an open set $E$ in the plane $\mathbb R^{2}$, is said to be real-analytic in $E$ if to every point $(s_{0}, t_{0})$ in there corresponds an expansion with ...
0
votes
2answers
41 views

Where does the series converges $\sum^\infty_{n=1} (-5)^n \sqrt[5]{\frac{(2n-1)!!}{(2n)!!}}x^{3n}$Is the solution OK?

So guys I want you to tell me is the solution OK. I'm terribly sorry for the not so detailed solution, but the writing in Latex is just too much for me. The calculations aren't that hard so it ...
2
votes
2answers
26 views

Partial sums for a power series

I'm having trouble finding the formula for the partial sums of this series, $$\sum_{n=1}^{\infty\:}{nz^n}$$ where $z$ is a complex number. I'm not looking for the answer just a nudge in the right ...
2
votes
1answer
15 views

Radius of convergence of series with alternating coefficients

I need to compute, with proof, the radius of convergence $R$ for the series $$\sum_{k=0}^\infty \left(2-(-1)^n\right)^n z^n,$$ which is similar to a geometric series, except that the terms alternate ...
1
vote
2answers
55 views

A question about convergence interval of power series

Could you give me some hint how to solve this problem: Suppose $a_n$ is sequence defined as $a_1=\frac12,a_{n+1}=\frac12\left({a_n}^2+a_n\right)$. I managed to prove that $a_n$ is decreasing ...
0
votes
2answers
34 views

Power Series - Reference Request (?)

I'm not sure if I've tagged that correctly as a reference request or not, but I'm nearly done with Kenneth Ross's book Elementary Analysis, and one of the topic's that's caught my interest to learn ...
0
votes
2answers
76 views

What's the behavior of $\displaystyle\sum_{n=1}^\infty (z+\sqrt{5}+2i)^{n!}$ outside its radius of convergence?

I want to check the behavior of $$\displaystyle\sum_{n=1}^\infty (z+\sqrt{5}+2i)^{n!}$$ outside its radius of convergence. I've tried to use the ratio test as follows: ...
3
votes
1answer
41 views

Finding the power-series of $\frac{1}{(2-x)^2}$

I am going through some old Calculus-tasks in preparation for an upcoming exam, and a seemingly simple task is being stubborn with me. We are simply to find the power-series of the function ...
3
votes
0answers
19 views

A notion of transcendence degree for fields of formal power series?

There is an intuitive sense in which one would like to say that the field of formal Laurent series $k((z))$ over a field $k$ has "transcendence degree $1$". Of course, it doesn't have transcendence ...
1
vote
1answer
36 views

Moving Center of Power Series

Given a power series: $$\lim_{N\to\infty}\sum_{k=0}^N A_k (z-a)^k$$ I expand the powers: $$\lim_{N\to\infty}\sum_{l=0}^N(\sum_{k=l}^N A_k \binom{k}{l}(-1)^{k-l}a^{k-l})z^l$$ But here I face the ...
2
votes
1answer
54 views

Series convergence of $\frac{(-1)^n}{x^{2n+1}}$ [closed]

Does this series converge, and if so how would I prove it? I thought of using the ratio test but I'm not sure. The series is $$ \sum_{n=0}^\infty\frac{(-1)^n}{x^{2n+1}}. $$
1
vote
1answer
22 views

$\left(\frac{a_n}{n^k}\right)_n$ is bounded implies $\sum_{n=0}^\infty a_nz^n$ has a radius of convergence $\ge 1$

Let $$\left(\frac{a_n}{n^k}\right)_n\subset\mathbb{C}\;\;\;\;\;(k\in\mathbb{N})$$ be a boundet sequence. I want to show that the power series $$\sum_{n=0}^\infty a_nz^n\;\;\;\;\;(a_n,z\in\mathbb{C})$$ ...
0
votes
1answer
28 views

$|a_{n}| \leq C e^{-|n|} \implies \sum_{n\in \mathbb Z} a_{n} e^{in(x+iy)} $ converges absolutely for $|y|<1$?

Suppose $\{a_{n}\} \subset \mathbb C$ with $|a_{n}| \leq C e^{-|n|}, n\in \mathbb Z$ and fix $C >0.$ My Question is: How to show the series, $$\sum_{n\in \mathbb Z} a_{n} e^{in (x+iy)}; (x, ...
0
votes
1answer
27 views

Convergence of a Power series

Consider the power series $\sum^{\infty}_{n=0} a_nx^n$. It is fairly easy to impose conditions on the value of $x$, so as to make the series convergent. However, I was wondering if it is possible to ...
1
vote
1answer
37 views

Question about a power series

For what value of $x$ does the series $$\sum_{}^{}\dfrac{(1+x)^n}{n(n-1)}$$ converge? Show that on a certain range of $x$ it determines a differentiable function whose derivative is $\log(-x)$. ...
1
vote
1answer
41 views

Root Test and Ratio Test

$$\sum_{n=1}^{\infty}\left(\dfrac{1}{2^n}\right)e^{(-1)^n\sqrt{n}}$$ How do I do the root test for this series? I know that the root test works and that the ratio test does not but how do I show ...
0
votes
2answers
69 views

Formal power series problem

So also have this differential equation: $$A''(z) + 4 A(z) = 0$$ With $A(z)$ stand for this classic formal power series $$A(z) = a_0 + a_1 z + ....$$ I need to show that the ...
2
votes
1answer
53 views

Power series - Calculate radius of convergence

Let $$\sum {n\over{n+1}} \cdot \left({{2x+1} \over x}\right)^n$$ I was asked to calculate the radius of convergence. We can write the series as: $$\sum {n\over {n+1}}\cdot \left(2+{1\over ...
1
vote
0answers
43 views

A theoretical question regarding Frobenius method

The following is a theoretical question regarding Frobenius method. Let $b(x),c(x)$ be real functions analytic at 0. Let $b(x)=\sum_{i=0}^\infty b_ix^i, c(x)=\sum_{i=0}^\infty c_ix^i$ on $(-R,R)$. ...
1
vote
2answers
21 views

Showing power series converges absolutely

Show that if the sequence ${a_n}$ is bounded then the power series $\sum a_nx^n$ converges absolutely for $|x|<1$. I haven't the slightest idea how to prove this. Does anyone have any thoughts on ...
1
vote
0answers
39 views

Integration through power series

Use series to estimate the value of the following function correct to $2$ decimal places: $$\int_0^1\sqrt{1+x^4}\mathrm dx.$$ I tried to express the function as a maclaurin series but I do not know ...
0
votes
0answers
38 views

Order of differentiation on a power series

I encountered something strange to me just now. Say we have $$f(x)=\ln(1+x^3)$$ Now, I want to find the power series expansion for $f'(t^2)$. I get two different answers for when I take the ...
1
vote
1answer
41 views

Power Series Solution to Differential Equation

The equation is $$y'' - xy' + y = 0$$ So far I have the recurrence relation - $$a_{n+2} = \dfrac{(n-1)a_n}{(n+1)(n+2)} $$ From this - $a_2 = \dfrac{-a_0}{2!}$ $a_3 = 0$ $a_4 = ...