MAT187 L12 - Integrals with Taylor Series

#MAT187 PCE
For certain functions that don't have an elementary antiderivative (i.e., those that cannot be found using the standard differentiation rules that we've learnt so far) we can use a Taylor Series to find a "good" approximation of the integral.

Example: $\int_{0}^{1} e^{- x^{2} / 2} d x$

This function does not have an elementary antiderivative. According to the PCE , this does not mean that the function is difficult to integrate conventionally, it means that it cannot, by anyone in the universe, be integrated with an explicit expression. Interesting choice to make such a strong and confident claim in a mathematics course without any proof, but not an unexpected one given Siefken's highly irregular teaching style. So, we can integrate a Taylor Series to approximate the value of the definite integral.
Recall from the previous lecture that, for $x \in R$ , the Taylor series is of $e^{x}$ is:

e^{x} = \sum_{n = 0}^{\infty} \frac{x^{n}}{n!}

To get the Taylor Series for the function $e^{x^{2}}$ , we can simply replace $x$ with $- x^{2}$ .^[1]

\begin{aligned} e^{- x^{2}} & = \sum_{n = 0}^{\infty} \frac{(- x^{2})^{n}}{n!} \\ = \sum_{n = 0}^{\infty} (- 1)^{n} \frac{x^{2 n}}{n!} \end{aligned}

This results in the same radius of convergence so it is still valid. (because $e^{x}$ converges $\forall x \in R$ )

Thus, we can integrate the Taylor Series:

\begin{aligned} \int_{0}^{1} e^{- x^{2}} d x & = \int_{0}^{1} \sum_{n = 0}^{\infty} (- 1)^{n} \frac{x^{2 n}}{n!} d x \\ = \int_{0}^{1} (1 - \frac{x^{2}}{2^{1}} + \frac{x^{4}}{2^{2}} + \dots) d x \end{aligned}

The proof of this was conveniently left out of the PCE, possibly because Siefken thought it would be too elementary to include. Alas, I am stupid, so I wanted to look into it a bit more. The reason for why this replacement is possible is because the general property of a power series centred at $x = 0$ is: $f (x) = \sum_{n = 0}^{\infty} c_{n} x^{n}$ , and this is defined for any value of $x$ where the function still follows the same convergence, i.e. the radius of convergence (in L14-15) remains the same. For example, we can substitute $187$ into the Taylor series approximation $f (x)$ and, though it might be a really bad approximation, it is still "valid". Thus, if $x$ itself is a function, then the approximation remains valid, and thus, we can substitute it into the approximation. ↩︎