_{∞} e = ∑ (n!)^{-1} ^{ n=0}
Here n! stands for the factorial of n.
The number e is relevant because one can show that the exponential function exp(x) can be written as e^{x}; the exponential function is important because it is, up to multiplication by a scalar, the unique function which is its own derivative and is hence commonly used to model growth or decay processes.
The number e is known to be irrational and even transcendental. It features (along with a few other fundamental constants) in the most remarkable formula in the world.