Euler's Equality: e^pi*i + 1 = 0

e^x = 1 + x + x²/2! + x³/3! + x⁴/4! + x⁵/5! + x⁶/6!...
cos(x) = 1 - x²/2! + x⁴/4! - x⁶/6!...
sin(x) x - x³/3! + x⁵/5!...

e^x*i = 1 + (x*i) + (x*i)²/2! + (x*i)³/3! + (x*i)⁴/4! + (x*i)⁵/5! + (x*i)⁶/6!...
      = 1 + (x*i) + (x²*i²)/2! + (x³*i³)/3! + (x⁴*i⁴)/4! + (x⁵*i⁵)/5! + (x⁶*i⁶)/6!...
      = 1 + x*i - x²/2! - (x³*i)/3! + x⁴/4! + (x⁵*i)/5! - x⁶/6!...
      = (1 - x²/2! + x⁴/4! - x⁶/6!...) + i*(x - x³/3! + x⁵/5!...)
      = cos(x) + i*sin(x)

e^pi*i + 1 = cos(pi) + i*sin(pi) + 1
              = ^-1 + 0 + 1 = 0