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.
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for a function
times differentiable at
which isTaylor Polynomials
The Taylor series for a function
which has derivatives of all orders at a pointDefinition
is given by
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.
to be the sequenceAs the Taylor Series
We actually define the Taylor series for
The first few Taylor polynomials for
The Taylor series:
For a Taylor polynomial,
For a Taylor series,
you need derivatives of all orders.
We could find Taylor series for
for the Taylor series forNotation
Using sigma notation write
The Taylor series for
is called the Maclaurin series forTerminology
Maybe; try it for
Start with a Taylor polynomial.
Try using a large degree.
Can you see the pattern?
Start by finding theNow What?
in the Taylor series for
get successive expressions for
Look at a couple more:
The coefficient for the degree
in the Taylor series for
Finding a Taylor series
means finding the coefficients.
Degree n term
Coefficient of Degree n Term
atA Weird Example
It’s hard to see that it even exists, but
Thus the coefficients in the Taylor series for
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.
on the interval (2,6).
Use a Taylor polynomial atExample
Find a polynomial approximation to
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.
A power series at
is a function of the form
are the coefficients.
All the coefficients are 1.
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
In other words,
for a given value of x
means we can substituteTrue For All x...
That gives us
which is the Maclaurin series for that function.
Start with the Maclaurin series for
It converges toOne Last Example
It converges to the function on the whole real line.
Memorize the following series:
The Maclaurin series for 1/(1-x)
to find Taylor series for
functions obtained from the above via