# Convergence Proofs

Prove that if $\displaystyle\sum|a_n|$ converges, then $\displaystyle\sum a_n^2$ converges.

Pf: Because $0 \leq a_n^2 + |a_n| \leq |a_n^2 + a_n|$ for all n, the series $\displaystyle\sum_{n=1}^\infty (a_n^2 +|a_n|)$ converges by comparison with the convergent series $\displaystyle\sum_{n=1}^\infty |a_n^2 + a_n|$. Furthermore, because $a_n^2 = (a_n^2 + |a_n|)-|a_n|$ we can write $\displaystyle\sum_{n=1}^\infty a_n^2 = \displaystyle\sum_{n=1}^\infty (a_n^2 + |a_n|)-\displaystyle\sum_{n=1}^\infty |a_n|$ where both series on the right converge, which implies that $\sum a_n^2$ converges.