Power series interact nicely with other calculus concepts.

### Reading derivatives from Taylor series

Recall that if has derivatives of all orders at , then the Taylor series centered at for is On the other hand, suppose we give you a series, that we claim is the Taylor series for a function . Given just the series, you can quickly evaluate , , , …, and so on. Let’s see an example.

### Solving differential equations using power series

If we have a differential equation we can frequently use Taylor series to obtain an approximate solution, which will be (hopefully) converge on some interval.

In his study of optics, George Biddell Airy developed the so-called *Airy function*, a
function that solves the differential equation for initial conditions , and . As innocent
as this differential equation seems, it is impossible to find a closed form solution!
Nevertheless, Taylor series will rescue us.

To compute higher derivatives, simply differentiate the differential equation: We can immediately write down the first terms of the Maclaurin series

### Integration

Just as we can differentiate term by term, we can also integrate term by term. This allows us to approximate many functions where we cannot find a ‘‘closed-form’’ formula. Recall the following theorem:

- is continuous and differentiable on .
- , with radius of convergence .
- , with radius of convergence .

- The theorem states that differentiation and integration do not change the
radius of convergence. It does not state anything about the
*interval*of convergence. They are not always the same. - Notice how the summation for starts with . This is because the constant term of goes to .
- Differentiation and integration are simply calculated term-by-term using the power rule.

We’ll use this idea to investigate the function an important function in signal analysis.

is the antiderivative of with , so , and To find the radius of convergence, use the ratio test. Write with me

and for any fixed this limit is zero. Hence our series converges on , with radius of convergence .