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.

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