Past exam of the mathematics course of the University of Cambridge / 2019 / ia / Paper 1 / 11E / b

Table of contents
- Solution b

Solution

The arithmetic-geometric mean inequality gives

a_{n} a_{n + 1} \leq \frac{a _{n} + a _{n + 1}}{2} .

Thus convergence of

\sum_{n} a_{n}

implies convergence of

\sum_{n} a_{n} a_{n + 1}

by the comparison test for series.

The converse is false. Define

a_{2 n} = \frac{1}{n}, a_{2 n - 1} = \frac{1}{n ^{3}} .

The even terms make

\sum_{n} a_{n}

diverge, while

a_{2 n - 1} a_{2 n} = \frac{1}{n ^{2}}, a_{2 n} a_{2 n + 1} = \frac{1}{n ( n + 1 ) ^{3}} \leq \frac{1}{n ^{2}},

so the neighboring geometric-mean series converges.

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