Exercise 2.7.8

Question

Consider each of the following propositions. Provide short proofs for those that are true and counterexamples for any that are not.
  \begin{enumerate}[label=(\alph*)] 
  \item If $\sum a_{n}$ converges absolutely, then $\sum a_{n}^{2}$ also converges absolutely.
  \item If $\sum a_{n}$ converges and $\left(b_{n}\right)$ converges, then $\sum a_{n} b_{n}$ converges.
  \item If $\sum a_{n}$ converges conditionally, then $\sum n^{2} a_{n}$ diverges.
   \end{enumerate}

Ulisse Mini · Accepted Answer

\begin{enumerate}[label=(\alph*)] 
  \item True since $(a_n) 	o 0$ so eventually $a_n^2 \le |a_n|$ meaning $\sum a_n^2$ converges by a comparsion test with $\sum |a_n|$.
  \item False, let $a_n = (-1)^n/{\sqrt n}$ and $b_n = (-1)^n/{\sqrt n}$. $\sum a_n$ converges by the alternating series test, but $\sum a_nb_n = \sum 1/n$ diverges.
  \item True, suppose $(n^2a_n)$ converges, since $(n^2 a_n) 	o 0$ we have $|n^2 a_n| < 1$ for $n > N$, implying $|a_n| < 1/n^2$. But if $|a_n| < 1/n^2$ then a comparsion test with $1/n^2$ implies $a_n$ converges absolutely, contradicting the assumption that $a_n$ converges conditionally. Therefore $\sum n^2a_n$ must diverge.
   \end{enumerate}

Exercise 2.7.8

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Exercise 2.7.8

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