Skip to main content

Geometric Series

In Probability,

โˆ‘j=kNaj=akโˆ’aN+11โˆ’a\sum\limits_{j=k}^{N} a^j = {{a^k - a^{N+1} \over {1-a}}} ย ifย โˆฃaโˆฃ<1,ย โˆ‘j=kโˆžaj=limโกNโ†’โˆžโˆ‘j=kNaj=ak1โˆ’a~\text{if}~ |a| < 1, ~\sum\limits_{j=k}^{\infty} a^j = \lim_{N \to \infty} \sum\limits_{j=k}^{N} a^j = {{a^k \over {1-a}}} โˆ‘j=1โˆžaj=a1โˆ’a\sum\limits_{j=1}^{\infty} a^j = {a \over {1-a}} โˆ‘j=0โˆžaj=a1โˆ’a\sum\limits_{j=0}^{\infty} a^j = {a \over {1-a}} โˆ‘j=1โˆžjaj=a(1โˆ’a)2\sum\limits_{j=1}^{\infty} ja^j= {a \over {(1-a)^2}} โˆ‘j=0โˆžjaj=a(1โˆ’a)2\sum\limits_{j=0}^{\infty} ja^j= {a \over {(1-a)^2}} โˆ‘j=1โˆžj2aj=a+a2(1โˆ’a)2\sum\limits_{j=1}^{\infty} j^2 a^j= {{a + a^2} \over {(1-a)^2}} โˆ‘j=0โˆžj2aj=a+a2(1โˆ’a)2\sum\limits_{j=0}^{\infty} j^2 a^j= {{a + a^2} \over {(1-a)^2}}