1
vote
1answer
61 views

Does the series $\sum\limits_{n=1}^\infty\frac{\sin(n)n!}{n^n}$ converge?

$\sum\limits_{n=1}^\infty\frac{\sin(n)n!}{n^n}$ Please let me know how you did it. Thank you.
1
vote
1answer
28 views

Determine the value of r where the series converges

show that $$ \big(r\big)^{ln(n)} = \big(n\big)^{ln(r)} $$ Then determine the values of r (with r>0) for which the series $$ \sum_1^\infty (\big(r\big)^{ln(n)})$$ converges. r must be in what ...
0
votes
1answer
23 views

Absolute and conditional convergence of a series with $\sin(x)$

I have to explore absolute and conditional convergence of this function series I tried to find $a(n)$ and $a(n+1)$ terms of the series and then divide it and take a limit. But I've got nothing. ...
0
votes
3answers
48 views

Find a radius of convergence of power series

I have to Find a radius of convergence of this power series I' ve decided to use D'alambert indication: Looking for a limit i meet a problem with a factorial Please. help me finish this ...
0
votes
2answers
32 views

Convergence of complex power series question

I need some help to solve this problem and find the domain of convergence of the following power series: $$\displaystyle\sum_{n=0}^\infty(2^n+i^n)(z-2i)^n$$ Thank you!
1
vote
2answers
49 views

Power Series Representation…

I am having a hard time understanding how to proceed with this question... Find a power series representation for the function and determine it's radius of convergence $$ f(x)= x^2\ln(1+x^2) $$ How ...
1
vote
2answers
38 views

Simplifying ratio test with exponents $k+1$

Question: Find the interval and radius of convergence. $$\sum_{k=1}^\infty\frac{(x-1)^k(k^k)}{(k+1)^k} .$$ I applied the ratio test. ...
1
vote
1answer
15 views

Converting from radius of convergence to interval of convergence

Using the root test I have determined that $$\sum n^{-n} x^n$$ has a radius of convergence of infinity and $$\sum n^{n} x^n$$ has a radius of convergence of 0. Does this mean that the respective ...
1
vote
1answer
50 views

Showing the power series of $\cos(x)$ converges uniformly to $\cos(x)$ on every bounded interval

Show that the power series of $\cos(x)$ converges uniformly to $\cos(x)$ on every bounded interval. My attempt: The power series for $\cos(x)$ is $$\sum_{n=0}^{\infty} ...
3
votes
1answer
31 views

Power series with interval of convergence of $(-1,1]$?

Is there a power series with an interval of convergence of $(-1,1]$? Wouldn't the fact that absolute convergence implies regular convergence make such a function impossible to find?
0
votes
1answer
35 views

Find the radius of convergence of the power series

$\displaystyle\sum_{n=0}^{\infty}a_nz^n$, where $a_{2k+1} = 2^k$ and $a_{2k} = (1 + (1/k))^2$ for $k = 0, 1, 2, \dotsc$ I started off by doing the ratio test, but I know that the ration test is for ...
1
vote
1answer
33 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} = ...
1
vote
0answers
72 views

What does the convergence of a Dirichlet series tells us about the convergence of a power series?

If $D(s)=\displaystyle \sum_{k\geqslant 1} f(k)\, k^{s}$ converges for $\Re(s)\lt a$, what is the radius of convergence of $\displaystyle \sum_{k\geqslant 1}f(k)\, x^k$ $=T(x)$? Conversely, what ...
1
vote
1answer
18 views

Finding the Power Series of a Complex fuction.

Find a power series expression $\sum_{n=0}^\infty A_n z^n $ for $ \frac{1}{z^2-\sqrt2 z +2} $ I'm completely stuck on this question. I know how to manipulate power series but I've never had to find ...
0
votes
2answers
47 views

Find $\sum_{n=0}^\infty\frac{(a|x|)^n}{\frac{n}{2}!}$ where $\frac{n}{2}!=\Gamma(\frac{n}{2}+1)$

Find A=$\sum_{n=0}^\infty\frac{(a|x|)^n}{\frac{n}{2}!}$ where $\frac{n}{2}!=\Gamma(\frac{n}{2}+1)$. I know that A converges (I used the ratio test) but I can't work out what it converges to. ...
1
vote
0answers
38 views

Convergence set of power series

I am trying to find the convergence set of the power series: $\sum_{n=1}^\infty ln\big[1+\big(\dfrac{1}{n}\big)\big](x+2)^n$. So using the ratio test: $\lim_{n\to\infty} \dfrac{|a_{n+1}|}{|a_n|} = ...
0
votes
1answer
33 views

Help with Taylor series problem

I am using maple to plot the graphs of e^e^x versus its truncated Taylor series around 0. For small values of x, the two graphs converge nicely, but once x<-3, my Taylor series loses control. Here ...
-1
votes
1answer
50 views

Convergence, Divergence and Summability of this series

If f(x) is an infinitely differentiable function at x=0 and $f^{(n)}(0)$ is the nth derivative of the function f at zero, then does the series below converge or diverge? $\sum_{n=0}^{\infty} ...
1
vote
1answer
93 views

Taylor series of $\frac 1 {1+x^2}$

I have to construct the Taylor series of $$\frac 1 {1+x^2}$$ around $0$ and $1$ and analyze the convergence in both cases. Also (but this is a consequence of the previous series) I have to construct ...
0
votes
1answer
56 views

Does $\sum_{n=0}^\infty\frac{a^n}{\frac{n}{2}!}x^n$ converge?

And if so, what is the radius of convergence of $x$? I am inclined to think it converges absolutely for all $x$ but I can't prove it. I have tried using an adaptation of the ratio test: ...
3
votes
3answers
71 views

Radius of convergence of the series $\displaystyle\sum\limits_{n=0}^\infty \frac{n!\,z^{2n}}{(1+n^2)^n}$

I am doing the following problem and would like to know whether my answer is correct or not: Find the Radius of convergence for the complex series $\displaystyle\sum\limits_{n=1}^n ...
1
vote
0answers
40 views

Behaviour of $\sum_{n=1}^{\infty}\frac{z^{n}}{n}$ for $|z| = 1$ [duplicate]

How to show that the power series $$\sum_{n=1}^{\infty}\frac{z^{n}}{n}$$ ( which has radius of convergence $1$ ) converges in all points of $\partial D(0,1)$ except $z = 1$ ?
0
votes
1answer
32 views

Power series (representation) of given function

Well I'm wondering if below power series is the correct result of the function - wolfram alpha doesn't give anything like the result. The function $$\frac{x}{2x^2+1} = x\cdot \frac{1}{1- \left ( ...
-1
votes
2answers
47 views

Power series with radius of convergence 2 that diverges at both -2 and 2?

I'm looking for a real power series that has radius of convergence 2 but diverges at both 2 and -2. Any idea? Thank you!
1
vote
1answer
82 views

Finding a radius of convergence of power series

I have to find the radius of convergence of some power series but I find myself in trouble for three of them : the series are $\sum2^kx^{k!}$ $\sum\sinh(k)x^k$ $\sum\sin(k)x^k$. For the first ...
1
vote
2answers
43 views

convergence ratio of the serie $e^{xn}$

How can I determine the values of $x$ such that the series converge: $$\sum_{n=0}^\infty e^{xn}$$ I'm really lost in this problem, please help.
1
vote
2answers
129 views

Converge of the sum $\sum_{k=1}^{n} k x^k $

For what values ​​of x the sum converges and what is the limit when $n \rightarrow \infty$ $\sum_{k=1}^{n} k x^k $ My work: First i try to calculate the interval and radius of convergence of ...
2
votes
2answers
78 views

Expansion and convergence of $\sum_{m=1}^{\infty}\frac{\sin(2\pi n x^{1/m})}{ m}$

Consider the series: $$\sum_{m=1}^{\infty}\frac{\sin(2\pi n x^{1/m})}{ m}\;\;\;\;n\in\mathbb{N}$$ Other than formal manipulation of the Taylor series of the $\sin$ function, is there a way to expand ...
1
vote
2answers
92 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 ...
1
vote
2answers
61 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 $$R = \lim_{n \rightarrow \infty} \frac {\mid a_n \mid} {\mid a_{n+1} \mid} = 0$$ ...
1
vote
1answer
96 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, $x$ ...
2
votes
3answers
66 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
71 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 ...
2
votes
1answer
62 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 ...
1
vote
3answers
77 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
vote
1answer
51 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
vote
2answers
136 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. Intuitively, I believe the answer is $|x| < 1$, but I am not sure how I can show that using the Weierstrauss Majorant ...
0
votes
1answer
62 views

Determine the sequence of coefficients $(a_n)_{n\in\mathbb{N_0}}$ so that: $\sum_{n=0}^\infty a_nx^n = \frac{e^x}{1-x} $

Assignment: Determine the sequence of coefficients $(a_n)_{n\in\mathbb{N_0}}$ so that $$\sum_{n=0}^\infty a_nx^n = \frac{e^x}{1-x}\ , \forall x\in\mathbb{R}: |x| < 1. $$ What I've got so far ...
1
vote
1answer
50 views

Determine the radius of convergence of the power series $\sum_{k=1}^\infty \frac{1-(-2)^{(-k-1)}k!}{k!} (z-2)^k$.

I could use some help solving this one. Applying the nth-root or the ratio test didn't work out for me. $$\sum_{k=1}^\infty \frac{1-(-2)^{(-k-1)}k!}{k!} (z-2)^k$$ Hints are just as appreciated ...
0
votes
1answer
35 views

Complex power series (or not quite so?)

I'm stuck with this problem. Any hints are appreciated. It just says $$ \mbox{"For what values of}\ z\ \mbox{is}\quad \sum_{n = 0}^{\infty}\left(z \over 1+z\right)^{n}\quad \mbox{convergent ?} $$ ...
1
vote
2answers
71 views

Why is the circle of convergence for complex power series a circle (and not e.g. a square)?

Power-Series have an "circle of convergence". With real numbers this is an interval. Expanding this to complex numbers this becomes a circle. There are lots of book stating this, but I did not find ...
1
vote
0answers
49 views

Radius of convergence for Taylor series?!

Given is: $f(x) = \frac{\sin x}{x} $ I need the Taylor series in $a = 0$, so: $$T(x,0) = \frac{1}{x} \sum_{n=0}^\infty ((-1)^n* \frac{x^{2n+1}}{(2n+1)!} ) = \sum_{n=0}^\infty (-1)^n * ...
1
vote
0answers
38 views

Where on the border of convergence circle series converges and where diverges?

I have power series of $ \sum\limits_{k=2}^{\infty} (\ln k)^{\alpha} z^k$. Alpha is a parameter. I've found the radius of convergence. R = 1. If $alpha \geq 0$ then series diverges for z from boundary ...
1
vote
1answer
26 views

Convergence of “sliced” power series

Let $\phi(t)=\sum_{k=1}^\infty a_k t^k$, $x=t^m \in \mathbb{C}$ for some fixed $m\in \mathbb{N}$ be a convergent power series. I guess that $a_0=0$. For $r=0,\ldots,m-1$ and $k=mq+r$, why are the ...
1
vote
2answers
199 views

Find the Radius of Convergence of the Series $\sum a_{n}x^{n^{2}}$ Using $\sum a_{n}x^{n}$?

I want to show that $$\sum a_{n}x^{n^2}$$ has radius of convergence of 1, using the fact that the power series $\sum a_{n}x^{n}$ has radius of convergence $R>1$, where $R$ is a real number ...
0
votes
0answers
126 views

Proving Convergence of a Power Series With Partial Sums

Let $\sum_{0}^{\infty} a_n$ be a series and $s_n$ its sequence of partial sums. Suppose $\sum a_nx^x$ converges for $|x| <1$; let $f(x)$ be its sum. Then $\displaystyle \sum_{0}^{\infty}s_nx^n = ...
5
votes
5answers
105 views

What is this limit equal to:

What is the following limit equal to and how do I prove it? $$\lim_{x\to 0^+} \frac{1}{1-\cos(x^2)}\cdot \sum_{n=4}^\infty{n^5x^n} $$ I've tried l'hospital but it doesn't seem to help since I don't ...
0
votes
4answers
116 views

Show the the series $\sum_{n,m=1}^\infty 1/(n+m)!$ is absolutely convergent and find its sum.

Show the the series $$\sum_{n,m=1}^\infty \dfrac{1}{(n+m)!}$$ is absolutely convergent and find its sum. This comes from a chapter called interchange of limit operations. I tried using the ratio ...
1
vote
1answer
317 views

how to find Radius of convergence for $\sum_{n=0}^{\infty}z^{n!}$?

how to find Radius of convergence for $$\sum_{n=0}^{\infty}z^{n!}$$ ? It seems to me that R=1 but I dont know how
0
votes
0answers
20 views

function series. x+1 or x-1? does it matter?

I have to look for convergence radius in some examples. got that. But is there any difference between the power series is $ (x+1)^n $ or $(x-1)^n $ 1 is $x_0$ is guess. right. and if $ |x-x_0| < ...