Geometric Series

Dave's Math Tables: Geometric Summations

(Math | Calculus | Expansions | Series | Geometric)

Summation Expansion Convergence Comments

n-1
sum r ⁿ
n=0 = 1 + r + r ² + r ³ + .. + r ^n-1
(first n terms) for r not equals 1,
= ( 1 - r ⁿ ) / (1 - r)
for r = 1,
= nr Finite Geometric Series

sum (n=0..inf) r ⁿ = 1 + r + r ² + r ³ + ... for |r| < 1, converges to
1 / (1 - r)
for |r| >= 1
diverges Infinite Geometric Series

See also the Geometric Series Convergence in the Convergence Tests.

Summation	Expansion	Convergence	Comments
n-1 r ⁿ n=0	= 1 + r + r ² + r ³ + .. + r ^n-1 (first n terms)	for r 1, = ( 1 - r ⁿ ) / (1 - r) for r = 1, = nr	Finite Geometric Series
r ⁿ	= 1 + r + r ² + r ³ + ...	for \|r\| < 1, converges to 1 / (1 - r) for \|r\| >= 1 diverges	Infinite Geometric Series