Alternate series test :

Let ,the alternate series test and converge if it satisfies the following conditions.

(1). ,

(2). for all values of .

Definitions of Absolute and conditional convergence :

(1) is Absolutely convergent if converges.

(2) is Conditionally convergent if converges but diverges.

Therefore, An alternating series is absolutely convergent if the absolute value of the series converges.

It is conditionally convergent if its absolute value does not converge, but it still converges as an alternating series.

An alternating series is absolutely convergent if the absolute value of the series converges.

It is conditionally convergent if its absolute value does not converge, but it still converges as an alternating series.