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.
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 $\displaystyle 2\sqrt{a_ka_{k+1}}\le a_k+a_{k+1}?$