# Thread: Real Analysis - Series

1. ## Real Analysis - Series

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)?

2. Originally Posted by ajj86
a. Show that if Σan converges absolutely, then Σ(an)^2 also converges absolutely. Does this proposition hold without absolute convergence?
As $\displaystyle \sum a_n$ converges eventualy $\displaystyle a_n<1$, then $\displaystyle a_n^2<a_n$ and you get the partial sums of $\displaystyle \sum a_n^2$ 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 $\displaystyle a_n=(-1)^n n^{-1/2}$ this will give a conditionaly convergent series, but $\displaystyle a_n^2=n^{-1}$ and the corresponding series will diverge as its the harmonic series.

CB

3. Originally Posted by ajj86
b. If Σan converges and an >= 0, can we conclude anything about Σsqrt(an)?
consider $\displaystyle a_n=1/n^2$, then consider $\displaystyle a_n=1/n^4$

CB