Past exam of the mathematics course of the University of Cambridge / 2007 / ia / Paper 1 / 3F / Solution

The two-variable arithmetic-geometric mean inequality follows from a square:Hence $2\sqrt{ab}\le a+b$, with equality exactly when $a=b$.

For the series, apply that same inequality with $a=a_n$ and $b=n^{-2}$ to obtainThe comparison series converges: its first summand has finite sum by hypothesis, and the inverse-square series converges, for example by the telescoping estimate $n^{-2}\le 1/[n(n-1)]=1/(n-1)-1/n$ for $n\ge 2$. Thus the comparison test for series provesEquivalently, its positive partial sums are increasing and bounded above by $\frac{1}{2}\sum a_n+\frac{1}{2}\sum n^{-2}$, so the bounded monotone sequence theorem proves convergence directly. This is the case $p=1$ of weighted square roots of a summable sequence.