Home  18.013A  Chapter 2  Section 2.1 


Figure out the series for exp x and prove it to be so.
Solution:
The power series expansion of exp x about 0 has the form
exp x = a_{0} + a_{1} x + a_{2} x^{2} + ...
When x is near 0, exp x is near 1. This implies a_{0} = 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 a_{1} = a_{0}, 2a_{2} = a_{1}, 3a_{3} = a_{2}, 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 ja_{j} = a_{j1}.
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
