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1.31 Geometric sequences

Geometric Sequence is (1,r,r2,r3,…)(1,r,r^{2},r^{3},\dots). It has the special property that the ratio of successive terms is always rr; that is,

new termold term=r.{{\hbox{new term}}\over{\hbox{old term}}}=r.

Proposition

(i) If r>1r>1, then rn→∞r^{n}\rightarrow\infty as n→∞n\rightarrow\infty and the sequence diverges to infinity.

(ii) If -1<r<1-1<r<1, then rn→0r^{n}\rightarrow 0 as n→∞n\rightarrow\infty and the sequence converges to zero.

(iii) For r=-1r=-1, the sequence (1,-1,1,-1,…)(1,-1,1,-1,\dots) oscillates and does not converge.