1. ## more convergence help

Suppose that $\displaystyle \sum$$\displaystyle a_{k} from k=1 to infinity is a convergent series of positive numbers. Prove that the series \displaystyle \sum$$\displaystyle \sqrt{a_{k}a_{k+1}}$ from k=1 to infinity converges.

2. Originally Posted by friday616
Suppose that $\displaystyle \sum$$\displaystyle a_{k} from k=1 to infinity is a convergent series of positive numbers. Prove that the series \displaystyle \sum$$\displaystyle \sqrt{a_{k}a_{k+1}}$ from k=1 to infinity converges.
Recall that $\displaystyle 2ab\le a^2+b^2$.

Does it follow that $\displaystyle 2\sqrt{a_ka_{k+1}}\le a_k+a_{k+1}?$