Figure out the series for exp x and prove it to be so.
The power series expansion of exp x about 0 has the form
exp x = a0 + a1 x + a2 x2 + ...
When x is near 0, exp x is near 1. This implies a0 = 1.
The derivative of exp x is itself and so is also near 1 when x is near 0.
Differentiating the series we find
This allows us to identify a1 = a0, 2a2 = a1, 3a3 = a2, from the fact that the coefficients of each power of x must be the same here and in the previous expression for exp x and in general jaj = aj-1.
This allows us to identify We conclude that the series for exp x is the sum from j = 0 to infinity of , which we write as