daozaich comments on Into the Kiln: Insights from Tao’s ‘Analysis I’

[daozaich]

What was initially counterintuitive is that even though $a_n\to 0$, the series doesn’t converge.

This becomes much less counterintuitive if you instead ask: How would you construct a sequence $a_n\to 0$ with divergent series?

Obviously, take a divergent series, e.g. ${\sum }_{n}1$, and then split the $n$th term $a_n=1$ into $n\times \frac{1}{n}$.