Activity 8.5.4

In this activity we will determine the Taylor series expansion for the famous error function, $$\begin{gathered} \erf(x) = \frac{2}{\sqrt{\pi}} \int_0^x e^{-t^2} \, dt \end{gathered}$$, that arises in probability theory and other important applications.

1. Use the Taylor series for $e^{x}$ to find the Taylor series for $e^{-t^2}$.

2. Next, evaluate the integral $\int_0^x e^{-t^2} \, dt$ by replacing $e^{-t^2}$ with its Taylor series.

3. Use your work in (b) to state the Taylor series for $\erf(x) = \frac{2}{\sqrt{\pi}} \int_0^x e^{-t^2} \, dt$

4. For what interval of $x$-values will the Taylor series for $\erf(x)$ converge? Why?

5. In probability theory, $\erf(x)$ is important because of its connection to the normal distribution, which is represented by a bell curve. Indeed, $\erf(x)$ represents the fraction of a normally distributed characteristic in a population that lies between $0$ and $x$. How can you use your result in (c) to estimate $\erf(0.5)$?