Real Analysis - Series

**ajj86** · November 2nd 2008, 04:51 PM

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)?

**CaptainBlack** · November 3rd 2008, 02:08 PM

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 $\sum a_n$ converges eventualy $a_n<1$ , then $a_n^2<a_n$ and you get the partial sums of $\sum a_n^2$ 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 $a_n=(-1)^n n^{-1/2}$ this will give a conditionaly convergent series, but $a_n^2=n^{-1}$ and the corresponding series will diverge as its the harmonic series.

CB

**CaptainBlack** · November 3rd 2008, 02:12 PM

Originally Posted by ajj86

b. If Σan converges and an >= 0, can we conclude anything about Σsqrt(an)?

consider $a_n=1/n^2$ , then consider $a_n=1/n^4$

CB

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