
Power series interact nicely with other calculus concepts.

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.

Let be the Taylor series for some function. What are the values of the function and the first eight derivatives when evaluated at zero?
You shouldn’t be using any derivative rules, instead, you should just ‘‘read-off’’ the derivatives from the series.

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.

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:

A few notes about the theorem above:

• 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.