(1) Decide if the following series converge or diverge. Explain your reasoning completely.  


(a)  

This is an alternating series   with   and .
Therefore, by the Alternating Series Test, it converges.


(b)  
Notice that     ,   and     is a divergent p-series.
Therefore, by the comparison test,    diverges.



(c)   
Using the Ratio Test,   .
It follows that the given series converges.



(d)  
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.



(e)

Therefore, by the Divergence Test, the series diverges.