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

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- Feb 12th 2010, 06:44 PMhebbySeries- 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 PMTheEmptySet
- Feb 12th 2010, 07:28 PMhebby
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 AMTheEmptySet
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.