Paper 1, Section II, F

Analysis I | Part IA, 2015

Let (an),(bn)\left(a_{n}\right),\left(b_{n}\right) be sequences of real numbers. Let Sn=j=1najS_{n}=\sum_{j=1}^{n} a_{j} and set S0=0S_{0}=0. Show that for any 1mn1 \leqslant m \leqslant n we have

j=mnajbj=SnbnSm1bm+j=mn1Sj(bjbj+1)\sum_{j=m}^{n} a_{j} b_{j}=S_{n} b_{n}-S_{m-1} b_{m}+\sum_{j=m}^{n-1} S_{j}\left(b_{j}-b_{j+1}\right)

Suppose that the series n1an\sum_{n \geqslant 1} a_{n} converges and that (bn)\left(b_{n}\right) is bounded and monotonic. Does n1anbn\sum_{n \geqslant 1} a_{n} b_{n} converge?

Assume again that n1an\sum_{n \geqslant 1} a_{n} converges. Does n1n1/nan\sum_{n \geqslant 1} n^{1 / n} a_{n} converge?

Justify your answers.

[You may use the fact that a sequence of real numbers converges if and only if it is a Cauchy sequence.]

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