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Geometric Series

In Probability,

j=kNaj=akaN+11a\sum\limits_{j=k}^{N} a^j = {{a^k - a^{N+1} \over {1-a}}}  if a<1, j=kaj=limNj=kNaj=ak1a~\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}}} j=1aj=a1a\sum\limits_{j=1}^{\infty} a^j = {a \over {1-a}} j=0aj=a1a\sum\limits_{j=0}^{\infty} a^j = {a \over {1-a}} j=1jaj=a(1a)2\sum\limits_{j=1}^{\infty} ja^j= {a \over {(1-a)^2}} j=0jaj=a(1a)2\sum\limits_{j=0}^{\infty} ja^j= {a \over {(1-a)^2}} j=1j2aj=a+a2(1a)2\sum\limits_{j=1}^{\infty} j^2 a^j= {{a + a^2} \over {(1-a)^2}} j=0j2aj=a+a2(1a)2\sum\limits_{j=0}^{\infty} j^2 a^j= {{a + a^2} \over {(1-a)^2}}