Question
Show that if $0<r<1,$ then $\lim _{n \rightarrow \infty} r^{n}=0 .$ Hint: Let $r=\frac{1}{1+p}$ where $p>0$. Then, by the Binomial Theorem, $r^{n}=\frac{1}{(1+p)^{n}}=\frac{1}{1+n p+n(n-1) \frac{p^{2}}{2}+\cdots+p^{n}}<\frac{1}{n p}$
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