Sorry, I mix up sum and integration,
the point is that
lim a_n * sum(b_n,1,inf) with lim a_n = 0 and b_n decreasing
is the same as
limit a_n * integrate(b_n,1,inf) = a_n * integrate(c_n,1,inf) with c_n
such that lim c_n /b_n = 1.
This is a version (or the proof is similar) of the well known p-integral test
for convergence.
The correct answer for the limit is:
(%i3) integrate(1/sqrt(n+i),i,1,n);
Is n-1 positive, negative, or zero?p;
(%o3) 2*sqrt(2)*sqrt(n)-2*sqrt(n+1)
(%i4) limit(%/sqrt(n),n,inf);
(%o4) 2*sqrt(2)-2