# Paper 4, Section II, D

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

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

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

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

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

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

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

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

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