# Series and Sums

Cuthbert Nyack
Arithmetic Series S = a + (a+d) + (a+2d) + ... + [a + (n-1)d] = n[2a + (n-1)d]/2

Geometric Series S = a + ar + ar2 + ar3 + ... + arn-1 = a(1 - rn)/(1 - r).

For |r| < 1 ; S = a/(1 - r)

Arithmetico-Geometrical Series S = a + (a + d)r + (a + 2d)r2 + (a + 3d)r3 + ... + {a + (N - 1)d}rn-1
= {a - rN[a + (N -1)d]}/(1 - r) + {dr(1 - rN - 1)}/(1 - r)2.

S = r1 + 2 r2 + 3 r3 + ... + N rN = (r(1 - (1 + N)rN + N rN+1)) /(1 - r)2

Integer Series
S = 1 + 2 + 3 + ... + N = N(N + 1)/2

S = 12 + 22 + 32 + ... + N2 = N(N + 1)(2N + 1)/6

S = 13 + 23 + 33 + ... + N3 = N2(N + 1)2/4

Binomial Series
(1 + x)n = 1 + nx + n(n-1)x2/2! + n(n-1)(n-2)x3/3! + ...

(y + x)n = yn + nxyn-1 + n(n-1)x2yn-2/2! + n(n-1)(n-2)x3yn-3/3! + ...

(1 + x)-1 = 1 - x + x2 - x3 + ...

'e' is defined as the limit of (1 + 1/x)x as x tends to infinity.

ex = 1 + x + x2/2! + ...

ln(1 + x) = x - x2/2 + x3/3 - ... for x > -1 and < or = +1.

Taylor's Series
f(x + h) = f(x) + hf'(x) + h2f''(x)/2! + ...

Maclaurin's Series
f(x) = f(0) + xf'(0) + x2f''(0)/2! + ...