1. Section 9.4b
Radius of convergence

2. Definition: Absolute Convergence
Recall the Direct Comparison Test from last class…
To apply this test, the terms of the unknown series must be
nonnegative. This doesn’t limit the usefulness of this test,
because we can apply it to the absolute value of the series.
Definition: Absolute Convergence
If the series of absolute values converges, then
converges absolutely.
Thm: Absolute Convergence Implies Convergence
If converges, then converges.

3. has no negative terms, and it is term-by-term less than or equal to:
A quick example with this new rule…
Show that the given series converges for all x.
This series
has no negative terms, and it is term-by-term less than or equal to:
Which we know
converges to e.
Therefore, converges by direct comparison.
Since converges absolutely, it converges.

4. Also recall this theorem from last class…
There are three possibilities for
with respect to convergence:
1. There is a positive number R such that the series diverges
for but converges for The
series may or may not converge at either of the endpoints
and
2. The series converges for every x
3. The series converges at x = a and diverges elsewhere
The number R is the radius of convergence

5. Using the Ratio Test to find radius of convergence
Find the radius of convergence of the given series.
Check for absolute convergence using the Ratio Test:

6. Using the Ratio Test to find radius of convergence
Find the radius of convergence of the given series.
So the ratio of each term to the previous term:
Which we need to be less than one:
The series converges absolutely (and hence converges) on
this interval, and diverges when x < –10 and for x > 10.
The radius of convergence is 10

7. Practice Problems The radius of convergence is 1
Find the radius of convergence of the given power series.
This is a geometric series with
So it will only converge when
The radius of convergence is 1

8. Practice Problems The radius of convergence is 1/3
Find the radius of convergence of the given power series.
Ratio Test for absolute convergence:
The series converges for or
The radius of convergence is 1/3

9. Practice Problems 8 The radius of convergence is
Find the radius of convergence of the given power series.
Ratio Test for absolute convergence:
The series converges for all values of x. 8
The radius of convergence is

10. Practice Problems
Find the radius of convergence of the given power series.
Ratio Test for absolute convergence:

11. Practice Problems The radius of convergence is 4
Find the radius of convergence of the given power series.
Ratio Test for absolute convergence:
The series converges for or
The radius of convergence is 4

12. Practice Problems The radius of convergence is 0
Find the radius of convergence of the given power series.
Ratio Test for absolute convergence:
The series only converges for x = 4.
The radius of convergence is 0

13. Practice Problems Interval of convergence:
Find the interval of convergence of the given series and,
within this interval, the sum of the series as a function of x.
This is a geometric series with:
It converges when:
Interval of convergence:

14. Practice Problems
Find the interval of convergence of the given series and,
within this interval, the sum of the series as a function of x.
This is a geometric series with:
Sum

15. Practice Problems Interval of convergence:
Find the interval of convergence of the given series and,
within this interval, the sum of the series as a function of x.
This is a geometric series with:
It converges when:
Which is always!!!
Interval of convergence:
Sum