Let $a_k$ be a sequence of real numbers. Let $A_n$ be a sequence of partial sums of $\sum_{k\ =\ 1}^{\infty} a_k$.
If $\sum_{k\ =\ 1}^{\infty} a_k$ converges (i.e. $A_n$ converges), then
(a) $\lim_{k \to \infty}a_k=0$,
(b) $A_n$ is bounded.
In other words, (a) and (b) are necessary conditions for the convergence of a series.
Question: Does $\sum_{k\ =\ 1}^{\infty} a_k$ converge, if (a) and (b) are true? In other words, are conditions (a) and (b) sufficient for the convergence of a series? If they are not, could anyone provide a counterexample?
Some relevant facts:
If $a_k\geq 0$, then (b) implies $A_n$ is a bounded monotonically increasing sequence; therefore, $\sum_{k\ =\ 1}^{\infty} a_k$ converges.
If $\sum_{k\ =\ 1}^{\infty} a_k$ is an alternating series, then it is convergent.
