Given $\sum a_n $ convergent, are $\sum n^{1/n} a_n$ and $\sum a_n/({1+ |a_n|})$ convergent?

Question

Let $\sum a_n $ be a convergent series. Is

1. $\sum n^{1/n} a_n$ convergent?
2. $\sum a_n/({1+ |a_n|})$ convergent?

If not, provide examples in the contrary.

I feel like 1 is maybe convergent, however I cannot seem to prove it or come up with an example to contradict. Please help.

Answer 1

Since there is no assumption that $a_n$ is non-negative, one must be careful when jumping to the conclusion.

1. Notice that $n^{1/n}$ is monotone and bounded on $n\geq 3$. So we can apply Abel's test to show that $\sum n^{1/n} a_n$ is convergent.

2. See this answer for a counter-example. A nuking-a-fly argument is to utilize the following theorem:

   Theorem. Let $f : \mathbb{R} \to \mathbb{R}$ satisfy the following property:

   $$ \forall (a_n) \in \mathbb{R}^{\mathbb{N}} \ : \quad \sum a_n \text{ converges} \quad \Rightarrow \quad \sum f(a_n) \text{ converges}. $$

   Then there exist $\delta > 0$ and $c \in \mathbb{R}$ such that $f(x) = cx$ on $(-\delta, \delta)$.