How can one determine if a convergent series raised to a power is also convergent? Forgive me if this is elementary, but my analysis is quite rusty and I'm struggling to get back up to speed. Given a convergent series $\sum_{i=1}^\infty|x_i|^p$, does it follow that $(\sum_{i=1}^\infty|x_i|^p)^\frac{1}{p}$ also converges? If so, how do they relate, i.e. is one always greater than (or equal) to the other? 

The sum is just a number (if it converges).  Call it S.  If S>0 and $p \ne 0$ you can calculate $S^{(1/p)}$
Maybe what you want to ask if if $\sum_{i=1}^\infty|x_i|$ converges does $\sum_{i=1}^\infty|x_i|^p$ converge?  It will if $p \ge 1$ as the $x_i$ are going to zero and raising them to a power greater than 1 will decrease them.
