For what values of r is the sequence {rn} convergent?

we know that limx → [infinity] ax = [infinity] for a > 1 and limx → [infinity] ax = 0 for 0 < a < 1. therefore, putting a = r and usingthis theorem, we have

lim n → [infinity] rn = [infinity] if r > 1 ,

lim n → [infinity] r^n = if 0 < r < 1.

it is obvious that

lim n → [infinity] 1n = and lim n → [infinity] 0n = .

if −1 < r < 0, then < |r| < , so

lim n → [infinity] |rn| = lim n → [infinity] |r|n =

and therefore limn → [infinity] rn = by this theorem. if r ≤ −1, then {rn} diverges as in this example. the figures show the graphs for various values of r.