Derivatives and Anti-Derivatives of Taylor Series

Since Taylor series are simply really, really long polynomials, you should differentiate and anti-differentiate them in exactly the same way as you would for polynomials.

This is helped by the fact that polynomials are everywhere defined, continuous, and differentiable any number of times. Polynomials are very well-behaved!

Now that we have refreshed our memory, just use the common derivative and integral rules, such as the fact that the derivative of a sum is the sum of the derivatives. Of course, the same holds for integrals.

Let us try an example. Suppose we are given the following Maclaurin series for the sine function: $$\sin x=\sum_{n=0}^{\infty}\frac{(-1)^n}{(2n+1)!}x^{2n+1}$$ Let us say that we would find the Maclaurin series for the cosine function.

We know the following: $$\frac{d}{dx}(\sin x)=\cos x$$ Thus, to find the Maclaurin series for cosine, simply differentiate the corresponding series for sine: $$\cos x=\frac{d}{dx}(\sin x)$$ $$\cos x=\frac{d}{dx}\left(\sum_{n=0}^{\infty}\frac{(-1)^n}{(2n+1)!}x^{2n+1}\right)$$ $$\cos x=\sum_{n=0}^{\infty}\frac{(-1)^n}{(2n+1)!}(2n+1)x^{(2n+1)-1}$$ $$\cos x=\sum_{n=0}^{\infty}\frac{(-1)^n}{(2n)!}x^{2n}$$ Thus, we have obtained quite easily the Maclaurin series for cosine using a simple differentiation.

Integration is simply the reverse, so the idea is the same. Here is an example.

Let us try to use series to integrate $f(x)=e^{-x^2}$, which does not have an elementary antiderivative.

We may use the following Maclaurin series for $e^x$: $$e^x=\sum_{n=0}^{\infty}\frac{x^n}{n!}$$ Replacing $x$ with $-x^2$ yields: $$e^{-x^2}=\sum_{n=0}^{\infty}\frac{(-1)^nx^{2n}}{n!}$$ Integrating gives us our result: $$\int e^{-x^2}dx=\sum_{n=0}^{\infty}\frac{(-1)^nx^{2n+1}}{n!(2n+1)} + C$$ Obviously, due to the infinite extent of the right side, this is of limited usefulness. However, taking the series to a finite number of terms can provide a very good approximation for the integral.

The differentiation and integration of Taylor series are extremely powerful tools for working with these series, since we are able to exploit the clean and obedient properties of polynomials. Keep this in mind whenever you must deal with series.