# Necessary and sufficient conditions for the convergence of a series

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.

How about $$1-\frac12-\frac12+\frac13+\frac13+\frac13-\frac14-\frac14-\frac14-\frac14+\cdots?$$ Here there are $k$ instances of $-(-1)^k/k$.