Converging series

**matty888** · April 12th 2008, 05:27 AM

Determine which of the following series,

, converge

i)

ii)

**Moo** · April 12th 2008, 05:36 AM

Hello,

For any x, $sin^2(x)+cos^2(x)=1$

So in ii), $a_n=1$ and $\sum_1^\infty 1$ is known to diverge

**angel.white** · April 12th 2008, 06:01 AM

Originally Posted by matty888

$a_n = \frac {n^2}{n!} = \frac {n}{(n-1)!}$

Using the Ratio test:
$\lim_{n->\infty} \left| \frac{a_{n+1}}{a_n}\right| = \lim_{n->\infty} \left| \frac{n+1}{n!}*\frac{(n-1)!}{n}\right|$

$=\lim_{n->\infty} \left| \frac{n+1}{n^2}\right|$

$=\lim_{n->\infty} \left| \frac{1+\frac 1n}{n}\right|$

$=0$

0 < 1 therefore the series is absolutely convergent and therefore convergent.

Originally Posted by Moo

$\sum_1^\infty 1$ is known to diverge

Not sure if that was a joke, but it was rather amusing ^_^

**Moo** · April 12th 2008, 06:08 AM

o.O

I was kinda serious, it diverges, doesn't it ?
The non-serious part was "known to", it depends on the level of the matty888 ^^

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