Geometric Summations

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Geometric Summations

(Math | Calculus | 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.