Question

If ∑ an with an > 0 is convergent, then is ∑ (an an+1)1/2 always convergent? Either prove it
or give a counterexample.

Answer 1

If we consider two convergent series , for which ∑(a(n))= A, and ∑(b(n)) = B, then the following also converge as indicated:
∑(a(n)+b(n)) = A + B
∑( (a(n)+a(n+1))/2) =& 0.5(∑(a(n) + ∑(a(n+1))) = 0.5∑(a(n))+0.5 ∑ (a(n+1)) .
Whereas the ∑(a(n)) converges absolutely (a(n)>0) , then ∑( (a(n)+a(n+1)/2)& converges too because at in right part of the equation there are two convergent series.