Harmonic series

$\sum\dfrac1n=1+\dfrac12+\dfrac13+\cdots$ — diverges, despite terms shrinking to $0$ .

The harmonic series $\sum_{n=1}^{\infty}\dfrac1n$ diverges, even though its terms $a_n=\frac1n$ shrink to $0$ — the classic proof groups terms into blocks ( $\frac12$ ; $\frac13+\frac14$ ; $\frac15+\frac16+\frac17+\frac18$ ; ...) each summing to AT LEAST $\frac12$ , so the partial sums grow without bound, just extremely slowly. It's the standard example proving the Divergence Test's converse is false.

Examples

The block $\frac13+\frac14\ge\frac14+\frac14=\frac12$ ; the block $\frac15+\frac16+\frac17+\frac18\ge4\cdot\frac18=\frac12$ — each doubling-sized block adds at least $\frac12$ forever.