Home  18.013A  Chapter 2  Section 2.1 


Figure out the series for $\mathrm{exp}x$ ; and prove it to be so.
Solution:
The power series expansion of $\mathrm{exp}x$ about 0 has the form
When x is near 0, $\mathrm{exp}x$ is near 1. This implies ${a}_{0}=1$ .
The derivative of $\mathrm{exp}x$ is itself and so is also near 1 when $x$ is near 0.
Differentiating the series we find
This allows us to identify ${a}_{1}={a}_{0},2{a}_{2}={a}_{1},3{a}_{3}={a}_{2}$ , from the fact that the coefficients of each power of $x$ must be the same here and in the previous expression for $\mathrm{exp}x$ and in general ${ja}_{j}={a}_{j1}$ .
This allows us to identify ${a}_{1}=1,{a}_{2}=\frac{1}{2}$ and in general ${a}_{j}=\frac{1}{j!}$ . We conclude that the series for $\mathrm{exp}x$ is the sum from $j=0$ to infinity of $\frac{{x}^{j}}{j!}$ , which we write as
