A number, approximately 2.718281828, that shows up absolutely everywhere in mathematics, including the limits of many infinite sums and continued fractions. It is the base of natural logarithms, which means that the area under the curve y=1/x between x=1 and x=e is exactly equal to one unit. e is important in probability and statistics, as well as number theory, analysis, and every other major branch of math. It is transcendental; it is related to pi by epi*i+1=0.

e is generally defined as
lim(n -> inf) (1 + 1/n)^n
Alternately you could write
Where the sum is clearly convergent for lim(n!/(n+1)!) = 0. The two definitions are of course equivalent.

Actually something a little more general is done . The WU below tries to explain how xr is defined when r is real. Note that for rational and integral r we have no problem. When r is irrational the situation becomes a little problematic. So this is what we do.

We start by defining
ex = sum(xn/n!)
Where the power series has clearly has a radius of convergence equal to infinite.

We then show that this power series expansion satisfies common properties of exponentials such as exp(a+b)=exp(a)*exp(b) etc. After this we define an inverse to the exponential function and call this inverse log(x). Finally we define
xr = er*log(x).
There. Thats it!

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