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Let [math]r[/math] be a real number, and consider the sequence [math]1, r, r^2, r^3, \ldots.[/math] Show that the sequence converges if and only if [math]-1 \lt r \leq 1[/math], and that

[[math]] \lim_{n\goesto\infty} r^n = \trilemma{0 & \mbox{if $-1 \lt r \lt 1$,}} {1 & \mbox{if \ltmath\gtr=1[[/math]]

,}} {\infty & \mbox{if [math]r \gt 1[/math].}} </math> What is the behavior of the sequence for [math]r=-1[/math] and for [math]r \lt -1[/math]? (Hint: Let [math]r^n = e^{n \ln r}[/math], for [math]r \gt 0[/math].)