1. Example Ex. For what values of x is the power series convergent?
Sol. By ratio test,
the power series absolutely converges when
and diverges when When we easily
see that it diverges when x=4 and converges when x=1. Thus
the power series converges for

2. Example Ex. Find the domain of the Bessel function defined by
Sol. By ratio test,
the power series absolutely converges for all x. In other words,
The domain of the Bessel function is

3. Characteristic of convergence
Theorem For a given power series
there are only three possibilities:
(i) The series converges only when x=a.
(ii) The series converges for all x.
(iii) There is a positive number R such that the series
converges if and diverges if
The number R is called the radius of convergence of the
power series. By convention, in case (i) the radius of
convergence is R=0, and in case (ii)

4. Characteristic of convergence
The interval of convergence of a power series is the
interval that consists of all x for which the series converges.
To find the interval of convergence, we need to determine
whether the series converges or diverges at endpoints |x-a|=R.
Ex. Find the radius of convergence and interval of convergence of the series
Sol
radius of convergence is 1/3.
At two endpoints: diverge at 5/3, converge at 7/3. (5/3,7/3]

5. Radius and interval of convergence
From the above example, we found that the ratio test or the
root test can be used to determine the radius of convergence.
Generally, by ratio test, if then
By root test, if then
Ex. Find the interval of convergence of the series
Sol when |x|=1/e, the general does not have limit zero, so diverge. (-1/e,1/e)

6. Representations of functions as power series
We know that the power series converges to
when –1<x<1. In other words, we can represent the function
as a power series
Ex. Express as the sum of a power series and find the interval of convergence.
Sol. Replacing x by in the last equation, we have

7. Example Ex. Find a power series representation for Sol.
The series converges when |-x/2|<1, that is |x|<2. So the
interval of convergence is (-2,2).
Question: find a power series representation for

8. Differentiation and integration
Theorem If the power series has radius of
convergence R>0, then the sum function
is differentiable on the interval and
(i)
(ii)
The above two series have same radius of convergence R.

9. Example
The above formula are called term-by-term differentiation and integration.
Ex. Express as a power series and find the radius of convergence.
Sol. Differentiating gives
By the theorem, the radius of convergence is same as the
original series, namely, R=1.

10. Example
Ex. Find a power series representation of
Sol.

11. Example Ex. Find a power series representation for
and its radius of convergence.
Sol.

12. Taylor series
Theorem If f has a power series representation (expansion) at a, that is, if
Then its coefficients are given by the formula
This is called the Taylor series of f at a (or about a)

13. Maclaurin series
The Taylor series of f at a=0 is called Maclaurin series
Ex. Find the Maclaurin series of the function and its radius of convergence.

14. Maclaurin series Ex. Find the Maclaurin series for sinx. Sol.
So the Maclaurin series is

15. Important Maclaurin series
Important Maclaurin series and their convergence interval

16. Example
Ex. Find the Maclaurin series of
Sol.

17. Multiplication of power series
Ex. Find the first 3 terms in the Maclaurin series for
Sol I. Find and the Maclaurin series is found.
Sol II. Multiplying the Maclaurin series of and sinx
collecting terms:

18. Division of power series
Ex. Find the first 3 terms in the Maclaurin series for
Sol I. Find and the Maclaurin series is found.
Sol II. Use long division

19. Application of power series
Ex. Find
Sol. Let then s=S(-1/2).
To find S(x), we rewrite it as

20. Exercise
Ex. Find
Sol.

21. Application of power series
Ex. Find by the Maclaurin series expansion.
Sol.
where in the last limit, we have used the fact that power
series are continuous functions.

22. Homework 25
Section 11.8: 10, 17, 24, 35
Section 11.9: 12, 18, 25, 38, 39
Section 11.10: 41, 47, 48