Past exam of the mathematics course of the University of Cambridge / 2024 / ia / Paper 1 / 3F / c

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.