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!