Is the Harmonic Series a Counterexample to a Convergent Series?

[alexmahone]

Give a counterexample to

$na_n\to 0,\ a_n\ge 0,\ a_n$ decreasing $\implies\sum a_n$ converges.

 

[Evgeny.Makarov]

Consider $a_n=\frac{1}{n\ln n}$.

 

[Greg Bernhardt]

A counterexample to this statement is the harmonic series, $\sum_{n=1}^{\infty}\frac{1}{n}$. This series satisfies the conditions given, as $\frac{n}{n} = 1 \to 0$ as $n\to\infty$, and the terms are all positive and decreasing. However, the harmonic series does not converge, as it is a well-known example of a divergent series. Therefore, the given statement is not always true.