Exercise 8

The fact that we can differentiate or integrate a Taylor series reveals other important ways we can think about functions such as $e^x$.

1. Perhaps the most important property of the function $h(x) = e^x$ is that $h'(x) = e^x$; that is, the function $e^x$ is its own derivative. Suppose that we didn't yet know the coefficients of the Taylor series expansion for $e^x$, so we just said $$\begin{gathered} e^x = a_0 + a_1 x + a_2 x^2 + a_3 x^3 + \cdots + a_n x^n + \cdots \end{gathered}$$. Let $x = 0$ in Equation; what does this tell us about the value of $a_0$?

2. Take the derivative of both sides of Equation and call your resulting equation for $e^x$ “Equation 2”. Why do Equation and Equation 2 together tell us that $a_1 = a_0$? Combine this observation with your conclusion in (b) and note that you now know the numerical value of both $a_0$ and $a_1$.

3. Why do Equation and Equation 2 together tell us that $a_2 = \frac{1}{2}a_1$?

4. Continue reasoning similarly to find the value of $a_3$, $a_4$, and $a_5$. What do you observe?