Series

Sum of geometric series

r < 1

Infinite sum of geometric series

S_n = a₀ / (1 - r)

Example

∑(n=0..∞) 1/2ⁿ = 1 + 1/2 + 1/4 + …

a₀ = 1/2⁰ ; r = 1/2

S_n = (1/2⁰) / (1 - 1/2) = 1 / (1/2) = 2

r > 1

Partial sum of geometric series

S_n = a₁ · ((rⁿ - 1)/(r - 1))

S_n = a₁ · ((1 - rⁿ)/(1 - r))

Example

1 + 2 + 4 + 8 + … + 1024

a₁ = 1 ; r = 2

n = 11 (There are 11 elements in total because the series is: 2⁰ + 2¹ + 2² + 2³ + … + 2¹⁰)

S_n = a₁ · ((rⁿ - 1)/(r - 1))

S_n = 1 · ((2¹¹ - 1)/(2 - 1)) = 2¹¹ - 1

Sum of arithmetic series

S_n = (n (a₁ + a_n)) / 2

Example

1 + 2 + 3 + … + N

S_n = (N (1 + N)) / 2 = (N (N + 1)) / 2

Sum of series

∑(i=1..n) c = c·n

∑(i=1..n) i = n(n+1) / 2

∑(i=1..n) i² = n(n+1)(2n+1) / 6

∑(i=1..n) i³ = n²(n+1)² / 4

∑(i=1..n) 1/i = 1