a. Show that if Σan converges absolutely, then Σ(an)^2 also converges absolutely. Does this proposition hold without absolute convergence?
b. If Σan converges and an >= 0, can we conclude anything about Σsqrt(an)?
As converges eventualy , then and you get the partial sums of eventualy form an increasing bounded sequence and hence the series converges.
If you only have conditional convergence this does not hold, a counter example will suffice to prove this: put this will give a conditionaly convergent series, but and the corresponding series will diverge as its the harmonic series.
CB