Existence of a series whose partial sums are bounded and $\lim_{k \to \infty} b_k = 0$, but does not converge?
Can one design such a series? Any ideas? Starting with the harmonic series, I reached nowhere. Again, the series is divergent, but some of the partial sums converge.
Best Answer
Here is an easy example:
$$(b_n) = \Big(\color{blue}{1}, \color{red}{-\frac{1}{2}, -\frac{1}{2}}, \underbrace{\color{blue}{\frac{1}{2^2}, \cdots, \frac{1}{2^2}}}_{2^2\text{-terms}}, \underbrace{\color{red}{-\frac{1}{2^3}, \cdots, -\frac{1}{2^3}}}_{2^3\text{-terms}}, \cdots \Big). $$
Its partial sum is bounded in $[0, 1]$ and the general term vanishes as $n\to\infty$, but nevertheless its sum does not converge.