Power Series. Dr. Dillon Calculus II Fall 1999. Recall the Taylor polynomial of degree. for a function. times differentiable at. which is. Taylor Polynomials. The Taylor series for a function. which has derivatives of all orders at a point. Definition. is given by. Compare.
Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.
is a (finite) sum
with a defined degree.
is an infinite sum, i.e.,
a sequence of partial sums.
Each member of the sequence of
partial sums is itself a Taylor
A Taylor series is a
sequence of Taylor polynomials.
Start with a Taylor polynomial.
Try using a large degree.
Can you see the pattern?
get successive expressions for
Look at a couple more:
Finding a Taylor series
means finding the coefficients.
Degree n term
Coefficient of Degree n Term
This Taylor series describes the function well
but only at one point, 0.
In cases like this, Taylor series aren’t good for much.
All of the usual suspects can be well represented by their Taylor series at all points where they are infinitely differentiable.
The Taylor series for all of our favorite functions converge to the functions at least on a decent sized interval, if not on the entire real line.
there are excellent polynomial approximations to the functions on intervals surrounding points
where the functions have derivatives
of all orders.
Use your technology to figure it out!
If the polynomial and the function
agree at the endpoints,
they agree at all the points between.
That’s a Big Theorem.
for some positive number
is the Taylor series for f at a.
is the Maclaurin series for
If a power series converges on an interval, we can
Memorize the following series:
The Maclaurin series for 1/(1-x)
to find Taylor series for
functions obtained from the above via