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

$\sum n^{1/n} a_n$ convergent?

$\sum a_n/({1+ |a_n|})$ 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.

What do you mean with the second series? I assume you do not mean $a_n/1$ since then you would just write $a_n$? — Václav Mordvinov, May 27 '18 at 11:04
For the first one we have by AM-GM that $n^{\frac{1}{n}}\leq{}(1+...+1)/n=1$ and thus it converges. — Μάρκος Καραμέρης, May 27 '18 at 11:06
See https://math.stackexchange.com/q/1025896/42969 for #1 and https://math.stackexchange.com/q/193486/42969 or https://math.stackexchange.com/q/348910/42969 for #2. — Martin R, May 27 '18 at 11:11
@MartinR, I did not realize that both are already answered before. Thank you for providing links. Do you think we should close the posting as duplicate? — Sangchul Lee, May 27 '18 at 11:19
@SangchulLee: I think so. (As a gold-badge holder you can add both other threads as duplicate targets.) — Martin R, May 27 '18 at 11:35
@MartinR, That functionality is exactly what I wondered. Thank you :) — Sangchul Lee, May 27 '18 at 11:38

score 2 · Accepted Answer · answered May 27 '18 at 11:15

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

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.
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 $(-\delta, \delta)$.

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

1 Answers1