Find an example of a series satisfying two properties

Question

Find a series $\sum_{n=1}^{+\infty} a_n$ such that:

• The general term is vanishing, that is $\lim_{n \rightarrow +\infty} a_n = 0$.
• The sequence of partial sums $s_n=\sum_{k=1}^{n} a_k $ is bounded .
• The series $\sum_{n=1}^{+\infty} a_n$ diverges.

Answer 1

Consider the sequence $(a_n)_{n\geq 0}$ made of "chunks" of increasing size. The $k$-th chunk has $2^k$ terms, each of them equal to $\frac{(-1)^k}{2^k}$.

The series $\sum_{n} a_n$ will have the properties you want.

Answer 2

$$ a=\{1,-1,\frac12,\frac12,-\frac12,-\frac12,\frac13,\frac13,\frac13,-\frac13,-\frac13,-\frac13,\frac14,\frac14,\frac14,\frac14,-\frac14,-\frac14,-\frac14,-\frac14,\ldots\} $$