## Abel Convergence Test

### Examples 4.2.18(a):

We already know that the series does not converge absolutely (why ?). As for convergence, let us verify the conditions for Abel's test:

First, let

and{ a_{n}} = { (-1)^{n}}

Then the sequence of partial sums of{ b_{n}} = { 1 / n }

*a*'s is clearly bounded (by what number ?), and the sequence

_{n}*{ b*is decreasing and convergent to zero. Hence, Abel's test applies, showing that the series converges.

_{n}}Therefore, the series converges conditionally.