Absolute and conditional convergence

 Objective: Determine whether a series is absolutely convergent or conditionally 

Theorem

Example. Determine whether the following series converges absolutely, conditionally or diverges.

Solution

Method

1) We start by calculating the series of the absolute value of the general term of the given series

2) We find a new series of which we study the convergence or divergence

3) If this series is convergent, the series of the absolute value of the general term is convergent. Therefore the given series converges absolutely

4) If this series diverges as in the example above, the series of the absolute value of the general term diverges. The series doesn't converge absolutely.  In this case we study the convergence of the given series. We conclude that the series converges conditionally.

Practice

Determine if the following series converges absolutely, conditionally or diverges.