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.:

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

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

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