Euler's Identity (Math Lair)

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Euler's identity is one of the most beautifully compact mathematical formulas. It is:

e^iπ = −1

or, more often

e^iπ + 1 = 0

The beauty of the identity lies in the fact that it succinctly expresses a relationship between five of the most fundamental mathematical constants:

e, the base of the natural logarithm
i, the square root of −1, the imaginary unit
π, the ratio of a circle's circumference to its diameter
1, the real unit and the multiplicative identity
0, the additive identity

This identity isn't too hard to prove, provided that you know elementary trigonometry as well as how to express the following functions as infinite series:

e^x = 1 + x + x²⁄2! + x³⁄3! + ...
sin x = x − x³⁄3! + x⁵⁄5! − ...
cos x = 1 − x²⁄2! + x⁴⁄4! − ...

Starting with the first one and letting x = iπ, we get:

e^iπ = 1 + iπ + i²π²⁄2! + i³π³⁄3! + i⁴π⁴⁄4! + ...
= 1 + iπ − π²⁄2! − iπ³⁄3! + π⁴⁄4! + ...
= (1 − π²⁄2! + π⁴⁄4! − ...) + i(π − π³⁄3! + π⁵⁄5! − ...)
= cos π + isin π
= −1 + 0
= −1