Suppose that $\sum$ $a_{k}$ from k=1 to infinity is a convergent series of positive numbers. Prove that the series $\sum$ $\sqrt{a_{k}a_{k+1}}$ from k=1 to infinity converges.
Suppose that $\sum$ $a_{k}$ from k=1 to infinity is a convergent series of positive numbers. Prove that the series $\sum$ $\sqrt{a_{k}a_{k+1}}$ from k=1 to infinity converges.
Recall that $2ab\le a^2+b^2$.
Does it follow that $2\sqrt{a_ka_{k+1}}\le a_k+a_{k+1}?$