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? 

You're just taking the $p$th root of the real number $s = \sum_{n=1}^{\infty} |x_{n}|^{p}$. So if $s \gt 1$ then $s \gt s^{\frac{1}{p}}$ and if $s \lt 1$ then $s \lt s^{\frac{1}{p}}$ finally, equality holds if $s = 1$.
