|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 series
of the form
is a variable and the
A power series is a function of for those for
which it converges.
, we have
But what good could this possibly be? Why is writing the simple
as the complicated series
of any value?
- 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
- For many functions, a power series is the best
explicit representation available.
, the Bessel function of order 0
arises as a solution to the differential equation
, and has the following power
This series is nice since it converges for all
prove this using the ratio test).
It is also one of the most explicit forms of