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
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 ...
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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 ...
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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 ...
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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 ...
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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 = ...
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0answers
42 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 ...
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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... ...
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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 ...
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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?
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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?
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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 ...
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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
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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
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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 ...
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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 ...
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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 ...
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2answers
77 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: ...
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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
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0answers
20 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 ...
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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 ...
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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 $$ \sum_{n=0}^\infty\frac{(-1)^n}{x^{2n+1}}. $$
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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 $$\sum_{n=0}^\infty a_nz^n\;\;\;\;\;(a_n,z\in\mathbb{C})$$ ...
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1answer
29 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, ...
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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 ...
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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 $\log(-x)$. ...
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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 ...
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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 ...
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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 ...
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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 $(-R,R)$. ...
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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 ...
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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 ...
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0answers
39 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 ...
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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 = ...
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1answer
21 views

Series Solution To Differential Equations - Need help with one step

Would someone kindly explain to me what the logic is behind one of the steps here: http://tutorial.math.lamar.edu/Classes/DE/SeriesSolutions.aspx In Example 1 - Following on from this sentence on ...
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1answer
46 views

Compute $\sum_{n=1}^\infty\frac{1}{(1-z^n)(1-z^{n+1})}z^{n-1}$ and show its uniform convergence

Given the power series $$P:=\displaystyle\sum_{n=1}^\infty\frac{1}{(1-z^n)(1-z^{n+1})}z^{n-1}$$ I want to show that $P$ converges uniformly in $\mathbb{C}$ and compute its limit. I've tried to ...
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0answers
19 views

Find the radius of convergence of the following

Here I am confused on which method to use, would it be Ratio Test or Hadamards Theorem. Any help would be appreciated.
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0answers
45 views

What is $f_\alpha(x) = \sum_{n\in \mathbb{N}} \frac{n^\alpha}{n!}x^n$?

I want to understand the function $$f_\alpha(x) = \sum_{n\in \mathbb{N}} \frac{n^\alpha}{n!}x^n, \ \ \ \forall x\in\mathbb{R},$$ for any possible real $\alpha\geq0$. I know that for $\alpha$ integer, ...
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3answers
53 views

Finding limit and sum of power series

Have such limit: $$ \lim \limits_{n \to \infty} \dfrac{1^{15}+3^{15}+ ...+ (2n-1)^{15}}{n^{16}} $$ But the sum $$ \sum_{n=1}^\infty (2n-1)^{15} $$ diverges. I think, that the answer is 0, ...
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1answer
28 views

Zeros of polynomials and power series

Consider $k_i \in \{-1,1\}$ for every $i \in \mathbb{N}$ and consider the family of polynomials $P$ of the form $$\mathop{\sum}\limits_{i=0}^n k_it^i;$$ and the family of power series $S$ of the form ...
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1answer
43 views

Radius of Convergence in Complex Analysis. [closed]

Following Questions are asked in previous years university exams. I'm preparing for the same exam to be held in next month. Please help me to solve these problems. I have no idea how to solve these ...
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1answer
67 views

Convergence Radius => Nonanalytic

Why is a function certainly nonanalytic at some point on the radius of convergence? I mean considering a power series around somewhere and if theres a power series expansion at every point on circle ...
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0answers
35 views

Find the interval of convergence for $\sum\limits_{n=0}^\infty\frac{(-1)^n}{n+1}\cdot x^{2n+2}$

I have a power series $$\sum\limits_{n=0}^\infty\frac{(-1)^n}{n+1}\cdot x^{2n+2}$$ and I need to find the interval of convergence. I'm not sure if I did this correctly. I said the interval of ...
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2answers
22 views

Finding power series expression

Hi could anyone help me with this problem. Use series to approximate the value of the following function to two decimal places. Integrate from 1 to 0 $\sqrt{1+x^4}$. I tried to differentiate the ...
2
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1answer
24 views

Power series solutions of differential equations, choosing x^n or x^(n+r)?

I cannot understand which one to use when solving differential equations by using power series solutions. For example in this question: Consider the following differential equation for $\alpha \in ...
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5answers
54 views

How to create alternating series with happening every two terms

I'm looking for a technique for creating alternating negatives and positives in a series. Specifically: when n=1, the answer is +, n=2 is +, n=3 is -, n=4 is -... etc. I have every other part of the ...
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0answers
76 views

expand a rational function in a power series

$$\frac{4-x}{(2-x)(1-x)^2}$$ Expand in ascending powers of x, stating when the expansion is valid; also write down the coefficient of $x^n $
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0answers
16 views

If power series converges to 0 $\forall$ $x \in (-R,R)$, then $a_n$ is $0$ for all $n$

Suppose that $$\sum\limits_{n=1}^\infty a_{n}x^{n}$$ converges for $x \in (-R,R)$. Show that if $f(x)=0$ for all $x \in (-R,R)$ then $a_n=0$ for all $n$. When I look at this , my guess is ...
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1answer
21 views

Determining the first few coefficients of the complex power series of $\frac{z + 1}{(z + 2) \cos z}

As the title states, I'm trying to find the coefficients $a_0$, $a_1$ and $a_2$ of the power series $\sum_{n = 0}^\infty a_n z^n$ around 0 of \begin{align*} \frac{z + 1}{(z + 2) \cos z}, \end{align*} ...
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1answer
28 views

Radius of Convergence of Sum of two Series.

Hi all, I know there are similar questions on here, but none deal with the fact of trying to prove that $T \geq min\{R,S\}$. Intuitively this doesn't make sense to me, If you have, ...
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2answers
46 views

Intervel of Convergence of a Power Series

Can anyone explain how to do this problem? I think you might be able to approach it with the ration test but I'm unsure. Any help is greatly appreciated! $$\sum_{n=0}^{\infty} \frac{(2x-3)^n}{n \ ...