# Power Series

 Final exam: Wednesday, March 22, 7-10pm in PCYNH 109. Bring ID! Quiz 4: This Friday Today: 11.8 Power Series, 11.9 Functions defined by power series Next: 11.10 Taylor and Maclaurin series

Recall that a polynomial is a function of the form

Polynomials are easy!!!
They are easy to integrate, differentiate, etc.:

Definition 6.5.1 (Power Series)   A power series is a series of the form

where is a variable and the are coefficients.

A power series is a function of for those for which it converges.

Example 6.5.2   Consider

When , i.e., , we have

But what good could this possibly be? Why is writing the simple function as the complicated series of any value?

1. Power series are relatively easy to work with. They are almost'' polynomials. E.g.,

where in the last step we re-indexed'' the series. Power series are only almost'' polynomials, since they don't stop; they can go on forever. More precisely, a power series is a limit of polynomials. But in many cases we can treat them like a polynomial. On the other hand, notice that

2. For many functions, a power series is the best explicit representation available.

Example 6.5.3   Consider , the Bessel function of order 0. It arises as a solution to the differential equation , and has the following power series expansion:

This series is nice since it converges for all (one can prove this using the ratio test). It is also one of the most explicit forms of .

Subsections
William Stein 2006-03-15