Series

Geometric Series is one where the ratio of successive terms in the
series is constant (r). a is the frist term.

a + ar + ar² + ar³ + ar⁴ + ...

Convergent, if –1>r>+1
Sum to infinity is S = a/(1 - r)

Sum of first n terms is
   S = a(1 – rⁿ)/(1 – r)

(1/2)+(1/4)+(1/8)+(1/16) ... = 1
(1/2)–(1/4)+(1/8)–(1/16) ... = 1/3

Arithmetic Series  is a sequence of numbers such that the
difference between the consecutive terms is constant.

a + (a+d) + (a+2d) + (a+3d) + ...

Sum of first n terms is
   S = (n/2)(a + an)
   S = (n/2)(2a+d(n–1))
nth term is
   an = a + nd
   where an is the nth term
   a is first term
   d is difference between terms

1 + 2 + 3 + 4 + ...
1 – 2 + 3 – 4 + ...


Sum the Integers from 1 to N inclusive
Sum = N(N+1)/2

from n to m inclusive
sum = (1/2)(m(m+1) – n(n+1))

Sum of first even integers
from 2 to N inclusive
Sum = N(N+1)
N = ((First Even + Last Even)/2) – 1
example, sum of 2+4+6+8+..+100
N = ((100+2)/2) – 1 = 50

Sum of first odd integers
from 1 to N inclusive
Sum = N²

Sum of first N digits separated by d
Sn = n/2(2a + (n-1)d)
a is first term
d is common diff


Taylor series
f(x) = f(a) + (f'(a)/1!)(x–a) + (f''(a)/2!)(x–a)^2 +
       (f'''(a)/3!)(x–a)^3 + ...

for sinx around 0
f(x) = 0 + (cos(0)/1)(x) – (sin(0)/2)(x)^2 –
       (cos(0)/6)(x)^3 + sin(0)/24)(x^4)...
f(x) = x – (x^3/3!) + (x^5/5!) – (x^7/7!)