Let $a_0$, $a_1$, $a_2$, $\ldots$ be a sequence of real numbers. Suppose that the sequence $(S_n)$ is bounded, where $S_n = \sum_{k=1}^n a_k$ and that $a_n \to 0$ for $n \to \infty$. Does this imply that $(S_n)$ converges?
[Math] Does a series with bounded partial sums converge if the summands go to $0$
analysissequences-and-series
Best Answer
This is not true. Consider the sequence $(a_n)$ given by $$ a_0, a_1, a_2, \ldots = 1, \underbrace{-\frac12, -\frac12}_2, \overbrace{\frac13, \frac13, \frac13}^3, \underbrace{-\frac14, -\frac14, -\frac14, -\frac14}_4, \overbrace{\frac15, \frac15, \frac15, \frac15, \frac15}^5, \ldots. $$ Then clearly $a_n \to 0$ for $n \to \infty$ and $S_n$ is bounded between $0$ and $1$. However, there are both infinitely many $n$ for which $S_n=0$ and infinitely many $n$ for which $S_n = 1$, so $(S_n)$ is not convergent.
If all $a_n$ are positive then $S_n$ must be convergent, because then $(S_n)$ is a bounded and increasing sequence.