Formulas

Geometric sums (finite)

1 + r + r 2 + r 3 + ⋅⋅⋅ + r n = 1 - r n + 1 1 - r ⁢

      for any real number `r!=1` and any positive integer `n`

Geometric series

1 1 - x = 1 + x + x 2 + x 3 + ⋅⋅⋅ + x k + ⋅⋅⋅ = ∑ k = 0 ∞ x k ⁢ ,    for   | x | < 1

Coefficients for Taylor polynomials and series

c k = f ( k ) ( 0 ) k ! for k = 0 ,   1 ,   2 ,   ... ⁢

Exponential series

e x = 1 + x + 1 2 ⁢ x 2 + ⋅⋅⋅ + 1 k ! ⁢ x k + ⋅⋅⋅ = ∑ k = 0 ∞ 1 k ! ⁢ x k

Sine series

sin ⁢     x = x - 1 6 ⁢ x 3 + 1 120 ⁢ x 5 + ⋅⋅⋅ + ( - 1 ) k ( 2 k + 1 ) ! ⁢ x 2 k + 1 + ⋅⋅⋅ = ∑ k = 0 ∞ ( - 1 ) k ( 2 k + 1 ) ! ⁢ x 2 k + 1

Cosine series

cos ⁢     x = 1 - 1 2 ⁢ x 2 + 1 24 ⁢ x 4 - ⋅⋅⋅ + ( - 1 ) k ( 2 k ) ! ⁢ x 2 k + ⋅⋅⋅ = ∑ k = 0 ∞ ( - 1 ) k ( 2 k ) ! ⁢ x 2 k

Error function series

erf ⁢     x = 2 π [ x - 1 3 ⁢ x 3 + 1 2 ! ⋅ 5 ⁢ x 5 - ⋅⋅⋅ + ( - 1 ) k k !   ( 2 k + 1 ) ⁢ x 2 k + 1 + ⋅⋅⋅ ] = 2 π ∑ k = 0 ∞ ( - 1 ) k k !   ( 2 k + 1 ) ⁢ x 2 k + 1

Natural logarithm series

ln   ( 1 + x ) = x - 1 2 ⁢ x 2 + 1 3 ⁢ x 3 - ⋅⋅⋅ + ( - 1 ) k + 1 k ⁢ x k + ⋅⋅⋅ = ∑ k = 0 ∞ ( - 1 ) k + 1 k ⁢ x k ,       for   | x | < 1

Arctangent series

arctan       x = x - 1 3 ⁢ x 3 + 1 5 ⁢ x 5 - ⋅⋅⋅ + ( - 1 ) k 2 k + 1 ⁢ x 2 k + 1 + ⋅⋅⋅ = ∑ k = 0 ∞ ( - 1 ) k 2 k + 1 ⁢ x 2 k + 1 ,       for   | x | ≤ 1

Binomial series

( 1 + x ) m = 1 + m ⁢ x + m ( m - 1 ) 2 ! x 2 + m ( m - 1 ) ( m - 2 ) 3 ! x 3 + ⋅⋅⋅ + m ( m - 1 ) ⋅⋅⋅ ( m - k + 1 ) k ! x k + ⋅⋅⋅

           = ∑ k = 0 ∞ ( m k ) ⁢ x k ,   for   | x | < 1

Harmonic series (divergent)

1 + 1 2 + 1 3 + ⋅⋅⋅ + 1 k + ⋅⋅⋅

Alternating harmonic series

1 - 1 2 + 1 3 - ⋅⋅⋅ + ( - 1 ) k k + ⋅⋅⋅ = ln ⁢   2

The Leibniz series

1 - 1 3 + 1 5 - ⋅⋅⋅ + ( - 1 ) k 2 k + 1 + ⋅⋅⋅ = π 4