Counterexample: If $\sum_{n=1}^{\infty}a_n=1$ and each $a_n\geq 0,$ then $\lim\limits_{n\to}na_n=0$

Question

I'm studying for an exam and I ran into these problems. I'm having a feeling that this is not true. Hence, I don't need to prove. I need to just provide a counterexample. However, the appropriate example is not just coming. Any help?

Prove or give a counterexample:

1. If $\sum_{n=1}^{\infty}a_n=1$ and each $a_n\geq 0,$ then $\lim\limits_{n\to\infty}na_n=0$

2. If $a_n\geq 0$ and $\sum_{n=1}^{\infty}a_n$ converges, then $\lim\limits_{n\to\infty}na_n=0$

Answer 1

Let $a_{n^2}=\dfrac{1}{n^2}$ and $a_n=0$ otherwise. Then $na_n$ does not tend to $0$, and the series is convergent.

The sum is $\dfrac{\pi^2}{6}$, but note you can multiply each term by the same constant to make the limit any nonzero number you wish.

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Note, however, that if $a_n$ is also nonincreasing, the situation is different: Series converges implies $\lim{n a_n} = 0$

Answer 2

Prescribe $(a_n)_n$ by: $a_{n}=n^{-1}$ if $n=2^k$ for some positive integer $k$ and $a_n=0$ otherwise.