# Series- Real Analysis

• Feb 12th 2010, 06:44 PM
hebby
Series- Real Analysis
Consider the series:

1-1/2 -1/3 +1/4 +1/5 -1/6 -1/7........where the signs come in pairs. Does the series converge or diverge?

I want to use the alternating series test....any ideas? how I would solve this.

Thanks
• Feb 12th 2010, 07:23 PM
TheEmptySet
Quote:

Originally Posted by hebby
Consider the series:

1-1/2 -1/3 +1/4 +1/5 -1/6 -1/7........where the signs come in pairs. Does the series converge or diverge?

I want to use the alternating series test....any ideas? how I would solve this.

Thanks

Try groupin together in pairs to get

$\displaystyle \left( \frac{2-1}{1\cdot2}\right) + \left( \frac{-4+3}{3\cdot 4}\right)+\left( \frac{6-5}{5\cdot 6}\right)+...+\frac{(-1)^{n+1}}{(2n-1)(2n)}+...$
• Feb 12th 2010, 07:28 PM
hebby
but in your case we get a plus and then a minus.....but the series is minus minus ...plus plus?
• Feb 13th 2010, 06:32 AM
TheEmptySet
Quote:

Originally Posted by hebby
but in your case we get a plus and then a minus.....but the series is minus minus ...plus plus?

As I said add them to gether in pairs. Write it out and you will see.

$\displaystyle a_1+a_2=1-\frac{1}{2}=\frac{1}{2}=b_1$
$\displaystyle a_3+a_4=-\frac{1}{3}+\frac{1}{4}=-\frac{1}{12}=b_2$
$\displaystyle a_5+a_6=\frac{1}{5}-\frac{1}{6}=\frac{1}{30}=b_3$

Note this is the series listed above I just gave the general term.

Since
$\displaystyle \sum a_n =\sum b_n$ we can look at the new series to draw any needed conclusions.