009C Sample Midterm 1, Problem 3 Detailed Solution

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Determine whether the following series converges absolutely,

conditionally or whether it diverges.

Be sure to justify your answers!

Background Information:  
1. A series    is absolutely convergent if
        the series    converges.
2. A series    is conditionally convergent if
        the series    diverges and the series    converges.


Step 1:  
First, we take the absolute value of the terms in the original series.
Step 2:  
This series is the harmonic series (or  -series with   ).
Thus, it diverges. Hence, the series
is not absolutely convergent.
Step 3:  
Now, we need to look back at the original series to see
if it conditionally converges.
we notice that this series is alternating.
First, we have
for all  
The sequence    is decreasing since
for all  
Therefore, the series     converges
by the Alternating Series Test.
Step 4:  
Since the series
converges but does not converge absolutely,
the series converges conditionally.

Final Answer:  
        conditionally convergent (by the p-test and the Alternating Series Test)

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