Question

Give an example of a series  ∑a_n such that  ∑a_n is not convergent but the sequence (a_n) converges to 0.

Answer 1

Consider the harmonic series

$$\displaystyle\sum_{i=1}^na_n=\displaystyle\sum_{i=1}^n\dfrac{1}{n}$$

The sequence $\{a_n\}$ defined by $a_n=\dfrac{1}{n}, n\geq1$ converges to $0.$

But the series $\displaystyle\sum_{i=1}^n\dfrac{1}{n}$ is divergent.