Series problems

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December 14th 2011, 10:26 PM
maoro

Series problems

#1
Find the radius of convergence and the interval of convergence of the series:

$\sum_{n=0}^{\infty}\frac{x^n}{nln(1+n)}$
I have attempted this problem as following
By root test
$lim_{n\to\infty}\sqrt[n]{|\frac{x^n}{nln(1+n)}|}$
$=lim_{n\to\infty}|x|\sqrt[n]{|\frac{1}{nln(1+n)}|}$
$=lim_{n\to\infty}|x|\frac{1}{|\sqrt[n]{ln(1+n)}|}$
$=\frac{|x|}{lim_{n\to\infty}\sqrt[n]{ln(1+n)}}$
$=\frac{|x|}{\sqrt[n]{lim_{n\to\infty}ln(1+n)}}$
$=\frac{|x|}{\sqrt[n]{lim_{n\to\infty}ln(1+n)}}$
$=\frac{|x|}{\sqrt[n]{ln(lim_{n\to\infty}(1+n))}}$
$=|x|$

When $x=1$ , $\sum_{n=0}^{\infty}\frac{1}{nln(1+n)}$ converges.
When $x=-1$ , $\sum_{n=0}^{\infty}\frac{(-1)^n}{nln(1+n)}$ converges.
Therefore, $R=1$ and $I=[-1,1]$

Is it correct?

#2
Assume that the positive series $\sum_{n=0}^{\infty} a_n$ is convergent.
Prove that the series $\sum_{n=0}^{\infty} ({a_n}^4+\frac{1}{n^2})$ is also convergent.
December 14th 2011, 10:30 PM
Chris11

Re: Series problems

For a power series, you do not have to prove convergance using root tests, because you know that the series converges within its radius of convergance, and this radius is defined whenever the limit lim a_n/a_n+1 exists. So compute the limit, then test the end points.
December 14th 2011, 10:57 PM
maoro

Re: Series problems

Quote:

Originally Posted by Chris11

For a power series, you do not have to prove convergance using root tests, because you know that the series converges within its radius of convergance, and this radius is defined whenever the limit lim a_n/a_n+1 exists. So compute the limit, then test the end points.

But is that true for using root test to prove it and is my computation true?
December 15th 2011, 02:19 AM
Prove It

Re: Series problems

Quote:

Originally Posted by Chris11

For a power series, you do not have to prove convergance using root tests, because you know that the series converges within its radius of convergance, and this radius is defined whenever the limit lim a_n/a_n+1 exists. So compute the limit, then test the end points.

Surely you mean the radius of convergence is defined wherever $\displaystyle \begin{align*} \lim_{n \to \infty}\left|\frac{a_{n+1}}{a_n}\right| < 1 \end{align*}$ ...
December 15th 2011, 03:17 AM
FernandoRevilla

Re: Series problems

Quote:

Originally Posted by maoro

#2Assume that the positive series $\sum_{n=0}^{\infty} a_n$ is convergent.
Prove that the series $\sum_{n=0}^{\infty} ({a_n}^4+\frac{1}{n^2})$ is also convergent.

As $a_n\to 0$ there exists $n_0\in\mathbb{N}$ such that $a_n^4\leq a_n$ for all $n\geq n_0$ . Hence, $0\leq a_n^4+\frac{1}{n^2}\leq a_n+\frac{1}{n^2}$ and $\sum_{n=1}^{\infty}\left(a_n+\frac{1}{n^2}\right)$ is convergent. So, ...

P.S. Surely there is a typo in your post: $n=1$ instead of $n=0$ .
December 15th 2011, 04:15 AM
maoro

Re: Series problems

Here is another problem is need help:
Use the ratio test to prove that the series $\sum_{n=1}^{\infty} \sin{\frac{\pi}{n}$ diverges
December 15th 2011, 04:52 AM
FernandoRevilla

Re: Series problems

$\lim_{n\to \infty}\frac{\sin (\pi /n)}{\pi /n}=1\neq 0$ , $\sin (\pi/n)\geq 0$ for all $n$ , and $\sum_{n=1}^{\infty}\pi/n$ is divergent, so the given series is divergent.
December 15th 2011, 04:55 AM
Prove It

Re: Series problems

Quote:

Originally Posted by maoro

Here is another problem is need help:
Use the ratio test to prove that the series $\sum_{n=1}^{\infty} \sin{\frac{\pi}{n}$ diverges

The ratio test is inconclusive because $\displaystyle \begin{align*} \lim_{n \to \infty} \left| \frac{\sin{ \left( \frac{\pi}{n + 1} \right) } }{ \sin{ \left( \frac{\pi}{n} \right) } } \right| = 1 \end{align*}$ ...
December 15th 2011, 04:58 AM
maoro

Re: Series problems

Quote:

Originally Posted by Prove It

The ratio test is inconclusive because $\displaystyle \begin{align*} \lim_{n \to \infty} \left| \frac{\sin{ \left( \frac{\pi}{n + 1} \right) } }{ \sin{ \left( \frac{\pi}{n} \right) } } \right| = 1 \end{align*}$ ...

ya, i also got this result and can't prove the desired divergence.
December 15th 2011, 08:56 AM
FernandoRevilla

Re: Series problems

Quote:

Originally Posted by maoro

ya, i also got this result and can't prove the desired divergence.

For that reason I applied the Limit Comparison Test: if $\sum a_n$ and $\sum b_n$ are series with positive terms and the limit $\lim_{n\to \infty}\frac{a_n}{b_n}$ exists and is nonzero, then either both $\sum a_n$ and $\sum b_n$ converge, or else both diverge.
December 15th 2011, 12:39 PM
Chris11

Re: Series problems

@proveit I was just telling him what the radius was.