Paper 4, Section II, D

Numbers and Sets | Part IA, 2017

Let (ak)k=1\left(a_{k}\right)_{k=1}^{\infty} be a sequence of real numbers.

(a) Define what it means for (ak)k=1\left(a_{k}\right)_{k=1}^{\infty} to converge. Define what it means for the series k=1ak\sum_{k=1}^{\infty} a_{k} to converge.

Show that if k=1ak\sum_{k=1}^{\infty} a_{k} converges, then (ak)k=1\left(a_{k}\right)_{k=1}^{\infty} converges to 0 .

If (ak)k=1\left(a_{k}\right)_{k=1}^{\infty} converges to aRa \in \mathbb{R}, show that

limn1nk=1nak=a\lim _{n \rightarrow \infty} \frac{1}{n} \sum_{k=1}^{n} a_{k}=a

(b) Suppose ak>0a_{k}>0 for every kk. Let un=k=1n(ak+1ak)u_{n}=\sum_{k=1}^{n}\left(a_{k}+\frac{1}{a_{k}}\right) and vn=k=1n(ak1ak)v_{n}=\sum_{k=1}^{n}\left(a_{k}-\frac{1}{a_{k}}\right).

Show that (un)n=1\left(u_{n}\right)_{n=1}^{\infty} does not converge.

Give an example of a sequence (ak)k=1\left(a_{k}\right)_{k=1}^{\infty} with ak>0a_{k}>0 and ak1a_{k} \neq 1 for every kk such that (vn)n=1\left(v_{n}\right)_{n=1}^{\infty} converges.

If (vn)n=1\left(v_{n}\right)_{n=1}^{\infty} converges, show that unn2\frac{u_{n}}{n} \rightarrow 2.

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