1.
Determine all values of a for which \(\displaystyle\sum_{n=1}^{\infty}\left(\dfrac{2}{a}\right)^n \) converges. I believe this to be (2,inf)
2.
Determine all values of a for which \(\displaystyle\sum_{n=1}^{\infty}\left(\dfrac{2}{n}\right)^a \) converges. I believe this to be (0,inf)
+505 1. Almost right; you forgot negative values 
(just as a reminder, the sum \(\displaystyle\sum_{n=1}^{\infty}x^n\) where x is a constant only converges when \(-1 by the ratio test.)
So the answer would be \((-\infty, -2)\bigcup(2, \infty)\)
2. The sum \(\displaystyle\sum_{n=1}^{\infty}\left(\dfrac{y}{n}\right)^a\), where y is a constant, only converges when \(a>1\), so the answer is \((1, \infty)\)
