Geometric Series -- From Wolfram MathWorld

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A geometric series sum_(k)a_k is a series for which the ratio of each two consecutive terms a_(k+1)/a_k is a constant function of the summation index k. The more general case of the ratio a rational function of the summation index k produces a series called a hypergeometric series.

For the simplest case of the ratio a_(k+1)/a_k=r equal to a constant r, the terms a_k are of the form a_k=a_0r^k. Letting a_0=1, the geometric sequence {a_k}_(k=0)^n with constant |r|<1 is given by

S_n=sum_(k=0)^na_k=sum_(k=0)^nr^k (1)

is given by

S_n=sum_(k=0)^nr^k=1+r+r^2+...+r^n. (2)

Multiplying both sides by r gives

rS_n=r+r^2+r^3+...+r^(n+1), (3)

and subtracting (3) from (2) then gives

(1-r)S_n=(1+r+r^2+...+r^n)-(r+r^2+r^3+...+r^(n+1)) (4)
=1-r^(n+1), (5)

so

S_n=sum_(k=0)^nr^k=(1-r^(n+1))/(1-r). (6)

For -1<r<1, the sum converges as n->infty,in which case

S=S_infty=sum_(k=0)^inftyr^k=1/(1-r) (7)

Similarly, if the sums are taken starting at k=1 instead of k=0,

sum_(k=1)^(n)r^k=(r(1-r^n))/(1-r) (8)
sum_(k=1)^(infty)r^k=r/(1-r), (9)

the latter of which is valid for |r|<1.

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