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Prove relationships between a given series

by RoRi

Let \{ a_n \} be a sequence of positive terms.

  1. Prove or give a counterexample: if \sum_{n=1}^{\infty} a_n diverges then \sum_{n=1}^{\infty} a_n^2 also diverges.
  2. Prove or give a counterexample: if \sum_{n=1}^{\infty} a_n^2 converges then \sum_{n=1}^{\infty} \frac{a_n}{n} also converges.
  1. Counterexample. Let a_n = \frac{1}{n}. Then

        \[ \sum_{n=1}^{\infty} \frac{1}{n} \]

    diverges. Furthermore, a_n^2 = \frac{1}{n^2} and

        \[ \sum_{n=1}^{\infty} \frac{1}{n^2} \]

    converges.

  2. Proof. We know from a previous exercise (Section 10.20, Exercise #51) we know

        \[ \sum a_n \quad \text{converges} \quad \implies \quad \sum \sqrt{a_n}n^{-p} \quad \text{converges} \]

    for p > \frac{1}{2}. So, for this case define b_n = a_n^2. Then \sum b_n = \sum a_n^2 converges by hypothesis. Furthermore, (since a_n > 0 so \sqrt{a_n^2} = a_n),

        \[ \sum \sqrt{b_n} n^{-1} = \sum \frac{a_n}{n}. \]

    Since this is the case p = 1, we have proved the statement. \qquad \blacksquare

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