1.32 Proof

(i) Let $a=r-1$, so $a>0$ and we can use the binomial theorem to expand

and hence $r^{n}\rightarrow\infty$ as $n\rightarrow\infty$.

(iii) The odd terms are all $1$; whereas the even terms are all $-1$.

(ii) First suppose that $0<r<1$, so $s=1/r>1$; then $s^{n}\rightarrow\infty$ as $n\rightarrow\infty$, and $r^{n}=1/s^{n}\rightarrow 0$ as $n\rightarrow\infty$.

Now when $-1<r\leq 0$, we let $t=-r;$ so $0\leq t<1$ and $r^{n}=(-1)^{n}t^{n}\rightarrow 0$ as $n\rightarrow\infty$.