]> Exercise 2.1

Exercise 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

d exp x d x = a 1 + 2 a 2 x + 3 a 3 x 2 +

This allows us to identify a 1 = a 0 , 2 a 2 = a 1 , 3 a 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 j a j = a j 1 .

This allows us to identify a 1 = 1 , a 2 = 1 2 and in general a j = 1 j ! . We conclude that the series for exp x is the sum from j = 0 to infinity of x j j ! , which we write as

exp x = j = 0 x j j !