May 8, 2002
(1) Decide if the following series converge or diverge. Explain your reasoning completely.
This is an alternating series with and .
Therefore, by the Alternating Series Test, it converges.
Notice that , and is a divergent p-series.
Therefore, by the comparison test, diverges.
Using the Ratio Test, .
It follows that the given series converges.
This series has positive and negative terms, but it's not alternating. Therefore, we check for absolute convergence. That involves investigating the series . Notice that < and it follows that converges by comparison with the convergent geometric series . Consequently, the original series converges absolutely, so it converges.
Therefore, by the Divergence Test, the series diverges.