Past exam of the mathematics course of the University of Cambridge / 2024 / ia / Paper 1 / 3F (Analysis I P∞)

Convergence of $\sum a_n$ implies $a_n\to 0$. Hence $0<a_n\le 1$ for all sufficiently large $n$, and then $a_n^2\le a_n$. The comparison test, with the finitely many initial terms treated separately, proves thatThis is the squares of a summable positive sequence lemma.

For every $N$, the Cauchy-Schwarz inequality givesThe right-hand side is bounded independently of $N$. Since the summands are positive, the partial sums increase to a finite limit, proving the geometric means of two summable positive sequences result.

Another application of the Cauchy-Schwarz inequality givesWhen $p>1/2$, the second factor is bounded because $2p>1$, so the series converges.

At the endpoint, take $a_1=1$ and, for $n\ge 2$,Then $\sum a_n$ converges, whilewhose series diverges. This proves both parts of weighted square roots of a summable sequence.