Absolute and Conditional Convergence

Definition A series   is called absolutely convergent if the series of absolute values   is convergent.

EXAMPLE 1 : The series

is absolutely convergent because

is a convergent p-series 

EXAMPLE 2:  We know that the alternating harmonic series

is convergent, but it is not absolutely convergent because the corresponding series of absolute values is 

which is the harmonic series  and is therefore divergent.

Definition A series o an is called conditionally convergent if it is convergent but not absolutely convergent.

Example 2 shows that the alternating harmonic series is conditionally convergent.

Thus it is possible for a series to be convergent but not absolutely convergent. However, the next theorem shows that absolute convergence implies convergence.

Theorem If a series  is absolutely convergent, then it is convergent.

EXAMPLE 3: Determine whether the series 

is convergent or divergent.

This series has both positive and negative terms, but it is not alternating.

(The first term is positive, the next three are negative, and the following three are positive:

the signs change irregularly.) We can apply the Comparison Test to the series of absolute values 

Since  for all  , we have

We know that  is convergent   and therefore  is convergent by the Comparison Test. Thus the given series is absolutely

convergent and therefore convergent

The following test is very useful in determining whether a given series is absolutely convergent