Problem 1124 (difficulty: 5/10)

True or false?

(1) If \(\displaystyle \sum\limits_{n=1}^\infty a_n\) is convergent, then \(\displaystyle \sum\limits_{n=1}^\infty (\root{n}\of{2}\cdot a_n)\) is also convergent.

(2) If \(\displaystyle \sum\limits_{n=1}^\infty a_n\) is divergent, then \(\displaystyle \sum\limits_{n=1}^\infty (\root{n}\of{2}\cdot a_n)\) is also divergent.

(3) If \(\displaystyle \sum\limits_{n=1}^\infty a_n\)is convergent, then \(\displaystyle \sum\limits_{n=1}^\infty \dfrac{a_n}{n}\) is also convergent.

(4) If \(\displaystyle \sum\limits_{n=1}^\infty a_n\) is divergent, then \(\displaystyle \sum\limits_{n=1}^\infty \dfrac{a_n}{n}\) is also divergent.

Give me another random problem!