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 = 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 ja_j = a_j-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