11.8. Power Series. 11.9. Representations of Functions as Power Series. Taylor and Maclaurin Series. 11.10. Power Series. A power series is a series of the form where x is a variable and the c n ’s are constants called the coefficients of the series.
Representations of Functions as Power Series
Taylor and Maclaurin Series
Graph of the Bessel function:
Approximating Functions with Polynomials
Example: Approximation of sin(x) near x = a
Brook Taylor was an accomplished musician and painter. He did research in a variety of areas, but is most famous for his development of ideas regarding infinite series.
1685 - 1731
Greg Kelly, Hanford High School, Richland, Washington
If we make , and the first, second, third and fourth derivatives the same, then we would have a pretty good approximation.
Suppose we wanted to find a fourth degree polynomial of the form:
that approximates the behavior of
If we plot both functions, we see that near zero the functions match very well!
This pattern occurs no matter what the original function was!
has the form:
(generated by f at )
(generated by f at )
If we want to center the series (and it’s graph) at zero, we get the Maclaurin Series:
Exercise 1: find the Taylor polynomial approximation at 0 (Maclaurin series) for:
The more terms we add, the better our approximation.
To find Factorial using the TI-83:
Exercise 2: find the Taylor polynomial approximation at 0 (Maclaurin series) for:
Rather than start from scratch, we can use the function that we already know:
Exercise 3:find the Taylor series for:
The 3rd order polynomial for is , but it is degree 2.
When referring to Taylor polynomials, we can talk about number of terms, order or degree.
This is a polynomial in 3 terms.
It is a 4th order Taylor polynomial, because it was found using the 4th derivative.
It is also a 4th degree polynomial, because x is raised to the 4th power.
The x3 term drops out when using the third derivative.
This is also the 2nd order polynomial.
1) Show that the Taylor series expansion of ex is:
2) Use the previous result to find the exact value of:
3) Use the fourth degree Taylor polynomial of cos(2x) to find the exact value of
Properties of Power Series:Convergence
Convergence of Power Series:
The Radius of Convergence for a power series is:
You must test each series that results at the endpoints of the interval separately for convergence.
Examples: The series is convergent on [-3,-1]
but the series is convergent on (-2,8].
Convergence of Taylor Series:
If f has a power series expansion centered at x = a, then the
power series is given by
And the series converges if and only if the Remainder satisfies:
Where: is the remainder at x, (with c between x and a).
Common Taylor Series: