Exercise 2.7.6

Question

Let's say that a series subverges if the sequence of partial sums contains a subsequence that converges. Consider this (invented) definition for a moment, and then decide which of the following statements are valid propositions about subvergent series:
  \begin{enumerate}[label=(\alph*)] 
  \item If $\left(a_{n}\right)$ is bounded, then $\sum a_{n}$ subverges.
  \item All convergent series are subvergent.
  \item If $\sum\left|a_{n}\right|$ subverges, then $\sum a_{n}$ subverges as well.
  \item If $\sum a_{n}$ subverges, then $\left(a_{n}\right)$ has a convergent subsequence.
   \end{enumerate}

Ulisse Mini · Accepted Answer

\begin{enumerate}[label=(\alph*)] 
    \item False, consider $a_n = 1$ then $s_n = n$ does not have a convergent subsequence.
    \item True, every subsequence converges to the same limit in fact.
    \item True, since $s_n = \sum_{k=1}^n |a_k|$ converges it is bounded $|s_n| \le M$, and since $t_n = \sum_{k=1}^n a_k$ is smaller $t_n \le s_n$ it is bounded $t_n \le M$ which by BW implies there exists a convergent subsequence $(t_{n_k})$.
    \item False, $a_n = (1, -1, 2, -2, \dots)$ has no convergent subsequence but the sum $s_n = \sum_{k=1}^n a_k$ has the subsequence $(s_{2n}) 	o 0$.
   \end{enumerate}

Exercise 2.7.6

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Exercise 2.7.6

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