Determine all values of $w$ for which $\sum\limits_{n=1}^{\infty}\left(\frac2n\right)^w$ converges.

Question

I need some help for these following connected questions in my calc workbook. The answer format is supposed to be in interval notation.
1) Determine all values of $w$ for which$\displaystyle\sum_{n=1}^{\infty}\left(\dfrac{2}{w}\right)^n$ converges.
I know the lower bound for w will have to be 2 for the fraction to be possible and i thought the upper bound would be inf but it was incorrect. I don't know where to go off on this?
2) Determine all values of $w$ for which $\displaystyle\sum_{n=1}^{\infty}\left(\dfrac{2}{n}\right)^w$ converges.
I also don't know where to start on this. Would w have to be negative numbers?
Thank you in advance!

Answer 1

Use the root test for 1). We have

$$ \lim \sqrt[n]{a_n} = \frac{2}{w} $$

Thus, as long as $w > 2$, we have convergence. For 2), notice that the sum is equivalent to

$$ 2^w \sum \frac{1}{n^w} $$

which converges as long as $w>1$ (p-series)