Originally Posted by

**osudude** Here is the problem:

If $\displaystyle \sum a_{n} $ with $\displaystyle a_{n}>0 $ is convergent, then is $\displaystyle \sum \sqrt{a_{n}a_{n+1}} $ always convergent? if not provide a counter example.

Now I don't think that it is also convergent. for example, $\displaystyle 2 \equiv \sqrt{2*2} < \sqrt {2*3} $. If I am right, what would be a counter example in terms of a? or anything else for that matter?

But if it is truly convergent, I know you have to approach this with the fact that a square root is non negative and you have to use a comparison test. but i just don't see how or why

$\displaystyle \sum \sqrt{a_{n}a_{n+1}} < \sum a_{n} $