Let $a_n > 0.$ When $\sum a_n$ converges $\sum \sqrt{a_n a_{n+1}}$ converges or not?
For, $$\frac{\sqrt{a_n a_{n+1}}}{a_n}=\frac {\sqrt{a_{n+1}}} {\sqrt{a_n}}$$
$\because$ By comparison test $\sum \sqrt{a_n}$ converges, $$\lim_{n\to\infty} \frac {\sqrt{a_{n+1}}} {\sqrt{a_n}}=1\neq0$$
Therefore by limit comparison test $\sum \sqrt{a_n a_{n+1}}$ converges.
Is this proof correct?
Corrections will be greatly appreciated...