2
votes
2answers
78 views

Convergence of the series $\sum\limits_{n\geqslant1}(2-x)(2-x^{1/2})(2-x^{1/3})\cdots(2-x^{1/n})$

If $x >0$, consider the series $\sum\limits_{n=1}^\infty a_n$, where $$a_{n}=(2-x)(2-x^{1/2})(2-x^{1/3})\cdots(2-x^{1/n}).$$ Is it convergent? This question is in my textbook. I don't know how to ...
0
votes
0answers
52 views

sum of the series $\sum_{k=0}^{\infty}(k+1)(x_n)^k.$

Let $x_n$ be a sequence of real numbers such that $x_n\in(0,1).$ Find the sum of the series $\sum_{k=0}^{\infty}(k+1)(x_n)^k.$ My answer is $\frac{(x_n)'}{(1-x_n)^2}.$ But the term $(x_n)'$ make me ...
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 ...
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. \ \ \ ...
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
1answer
50 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
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 ...
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 ...
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
2answers
67 views

Prove $f(x)=\sum_{k=1}^\infty \frac{1}{k!\sqrt{k}}x^k $ defines a continuous function on $\mathbb{R}$.

Prove $$f(x)=\sum_{k=1}^\infty \frac{1}{k!\sqrt{k}}x^k$$ defines a continuous function on $\mathbb{R}$. I think we can show that if $\sum_{k=1}^\infty \frac{1}{k!\sqrt{k}}x^k $ is uniformly ...
1
vote
2answers
80 views

radius of convergence $\displaystyle \sum_{n=1}^{\infty}n! x^{n!}$

I just wondering radius of convergence following series $$ \sum_{n=1}^{\infty}n! x^{n!} \\ $$ My 1st attempt is 'root test' $$ \sqrt[n!]{|a_{n!} |} =\sqrt[n!]{|n! |} =\sqrt[t]{t} \rightarrow 1 $$ So, ...
3
votes
1answer
65 views

Is there a generalization of the fundamental theorem of algebra for power series?

Given the similarity between polynomials and power series, I was wondering if there is any generalization of the fundamental theorem of algebra for power series. I understand that it doesn't make much ...
4
votes
1answer
44 views

Limit of ratio between a power series and a “subset” of the power series

$B$ is an infinite power series that converges everywhere, and $A$ is an infinite power series that converges everywhere which is composed only of terms found in $B$ - both have nonnegative real ...
4
votes
1answer
52 views

Some computation with a power series

This is an old qualifying exam problem that has me stumped. Suppose $$F(x):=\sum_{n=0}^{\infty}a_n x^n$$ converges in some neighborhood of the origin. I want to compute $$\sup\left\{\delta>0 : ...
0
votes
1answer
58 views

Question about infinitely many times differentiable function.

Could you please give me some hint how to solve this problem: Suppose $f(x)$ is infinitely many times differentiable function on R, $f(0)=f'(0)=f''(0)=0$. Prove : for all $A>0$ exists some ...
0
votes
1answer
31 views

Do I have mistakes in my calculations (power series, convergence)?

I'm not sure I got all of these problems right. I'd really appreciate any sort of feedback. For which $x \in \mathbb{R}$ do the following series converge? Problem 1 For ...
0
votes
0answers
64 views

Taylor theorem remainder term

I'm having trouble applying the formula for the remainder in the Taylor's theorem. From Wikipedia we know that for $f(x)=f(a)+f'(a)(x-a)+…\frac{f^{(n)}(a)}{n!}(x-a)^{n}+R$ the remainder $R$ in the ...
2
votes
1answer
41 views

Two questions about this solution /proof

Consider the following theorem: If $\sum_{n=0}^\infty a_n x^n $ converges for all $x \in (-R,R)$ then the differentiated series $\sum_{n=0}^\infty n a_n x^{n-1}$ converges for all $x \in (-R,R)$. ...
1
vote
0answers
37 views

Proof of pointwise convergence of derivative of power series

I proved: If $\sum_{n=0}^\infty a_n x^n$ converges for all $x \in (-R,R)$ then the differentiated series $\sum_{n=1}^\infty na_n x^{n-1}$ converges for all $x \in (-R,R)$. Please could somebody tell ...
3
votes
0answers
57 views

Proof of Abel's theorem

I tried to prove: If $g(x) = \sum_{n=0}^\infty a_n x^n$ is a power series that converges at $x= R > 0$ then it converges uniformly on $[0,R]$. Please can you check my proof? Let $\varepsilon ...
2
votes
3answers
20 views

Absolute sequence unbounded within radius of convergence

Let $R$ be the radius of convergence of the complex power series $a_nz^n$ with $0<R<\infty$. Show that if $|z|>R$, then the sequence $|a_nz^n|$ is unbounded. Trying by contradiction: So ...
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 ...
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
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 ...
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 ...
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 ...
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 ...
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 ...
0
votes
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, ...
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 ...
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 ...
0
votes
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 ...
2
votes
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, ...
3
votes
3answers
161 views

Help with radius of convergence of a power series.

I need to determine the radius of convergence of the series $\sum_{n=1}^\infty a_nx^n$, where $a_n=a^n+b^n$ and $a,b$ are real numbers. Not sure how to approach this one.
7
votes
3answers
262 views

Take 2: When/Why are these equal?

This didn't go right the first time, so I'm going to drastically rephrase the query. As per this previous question, I am wondering if the two series ...
3
votes
1answer
36 views

When are these series equal?

Suppose we have a power series $$\sum_{n=0}^\infty {a_nb_nx^n}$$ When is it true that the series obtained by eliminating $b_n$ is proportional to the original series? $$\sum_{n=0}^\infty ...
1
vote
1answer
47 views

How to justify, $\sum_{n=1}^{\infty} a_{n} x^{n} - \sum_{n=1}^{\infty}a_{n}y^{n}=\sum_{n=1}^{\infty} a_{n} (x^{n}-y^{n})$?

Let $\{a_{n}\}_{n\in \mathbb N} \subset \mathbb C$ so that the series, $\sum_{n=1}^{\infty} a_{n} x^{n},$ converges absolutely for all $x\in \mathbb R$ and we let $K_{1}$ be a compact subset of ...
2
votes
2answers
47 views

Is it true that, $|e^{x}-e^{y}|\leq C \cdot |x-y|$?

Define $f:\mathbb R \to \mathbb R$ such that $f(x)= e^{x}-1:= \sum_{n=1}^{\infty} \frac{x^{n}}{n!};$ for $x\in \mathbb R.$ My Question: Can we expect $|f(x)-f(y)|\leq |x-y| \cdot C;$ where $C$ is ...
0
votes
1answer
19 views

Radius of convergence of $\sum a_nx^n$ where $a_n = {k \choose n}$

Consider the power series $\sum a_n x^n$ where $$ a_n = {k \choose n} $$ for some $k$. What is the radius of convergence of this power series? I got one. Does that seem correct? I got that the ...
1
vote
1answer
46 views

Proving that the derivative of the integral of a power series equals the original power series

I've been thinking about the following recently: If we have a power series $f(x) = \sum_{n=0}^\infty a_n(x-c)^n$ and $F(x)=\sum_{n=0}^\infty \frac{a_n}{n+1}(x-c)^{n+1}$ where $F(x)$ is constructed by ...
1
vote
3answers
227 views

Find complicated Taylor Series

According to some software, the power series of the expression, $$\frac{1}{2} \sqrt{-1+\sqrt{1+8 x}}$$ around $x=0$ is $$\sqrt{x}-x^{3/2}+\mathcal{O}(x^{5/2}).$$ When I try to do it I find that I ...
0
votes
0answers
33 views

Reference for power series

I would need some references for power series, Taylors series of elementary functions, derivation and integration of power series, convergence of sequences of functions and series of functions. The ...
0
votes
1answer
26 views

Radius of convergence problem

I need a hint on this problem, been staring on the blackbord a long time now. Problem: Suppose $f(x) = \sum a_{n}(x-x_{0})^n$ has radius of convergence $R$ and $0 < r < R_{1} < R$. Show that ...
0
votes
0answers
28 views

derivatives of a composite function

There exists a 'closed-form' formula for the higher order derivatives of a composite function $f\circ g$ (it's called Faa di Bruno's formula but the formula itself is not my question). In their very ...
0
votes
1answer
58 views

Does a series always diverge if its sequence isn't a null-sequence?

I have the following series: $$\sum_{n = 1}^\infty 2^{n^2}z^n$$ The task is to give its radius of convergence. I solved that one using the root-test and came to the same answer. But the solution ...
1
vote
1answer
48 views

Radius and Interval of Convergence for Power Series

Find the radius and interval of convergence for the power series $\displaystyle{\sum_{k=1}^{\infty}} \frac{(x+3)^k}{k(6+(-1)^k)^k}$ I found that R=1 by calculating $\frac{1}{R} = ...
0
votes
1answer
50 views

Term wise differention

Consider $S(x) = \displaystyle{\sum_{k=1}^{\infty}} x^k k^2$. (a) Find an explicit formula for $S(x)$ on the interval $-1<x<1$ by repeated termwise differentiation of a geometric series. Be ...