Paper 4, Section II, 7E7 \mathbf{E}

Numbers and Sets | Part IA, 2007

Let xn(n=1,2,)x_{n}(n=1,2, \ldots) be real numbers.

What does it mean to say that the sequence (xn)n=1\left(x_{n}\right)_{n=1}^{\infty} converges?

What does it mean to say that the series n=1xn\sum_{n=1}^{\infty} x_{n} converges?

Show that if n=1xn\sum_{n=1}^{\infty} x_{n} is convergent, then xn0x_{n} \rightarrow 0. Show that the converse can be false.

Sequences of positive real numbers xn,yn(n1)x_{n}, y_{n}(n \geqslant 1) are given, such that the inequality

yn+1yn12min(xn,yn)y_{n+1} \leqslant y_{n}-\frac{1}{2} \min \left(x_{n}, y_{n}\right)

holds for all n1n \geqslant 1. Show that, if n=1xn\sum_{n=1}^{\infty} x_{n} diverges, then yn0y_{n} \rightarrow 0.

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