Consider a series Summationinfinityn1 an Suppose that summat

Consider a series Summation^infinity_n=1 a_n. Suppose that summation^infinity_n=1 a_n converges absolutely. Prove that summation^infinity_n=1 (a_n)^2 converges. Is the result from (a) still true if summation^infinity_n=1 a_n only converges conditionally? Either prow the claim or find a counterexample.

Solution

Proof: Let sn=ni=1a2n. I want to show that (sn) is a Cauchy sequence. So, let >0 and n>m, and consider,|snsm|=|(a21+a22+a2n)(a21+a22+a2m)|=|(am+1)2+(am+2)2++a2n|=|am+1|2+|am+2|2++|an|2Now, since am converges, the sequence (am) has limit 0. Therefore I can choose an N such that |am|</n for all m>N. So for all n>m>N, we have,|snsm|=|am+1|2+|am+2|2++|an|2<(n)2+(n)2++(n)2<(nm)2+(nm)2++(nm)2=Therefore (sn) is Cauchy and a2n converges. And since a2n=|a2n|, the series a2n is absolutely convergent.