Home | 18.013A | Chapter 2 | Section 2.1 | | Tools Glossary Index Up Previous Next

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 !

Up Previous Next