1. Section 11.5 – Testing for Convergence at Endpoints

2. There are lots of other tests we can use to test for convergence.
The Integral Test
This is useful in making decisions on series such as:

3. Another nice test: The Alternating Series Test
Harmonic is a special case of this.
Another nice test: The Alternating Series Test
We can use all of these facts to test the endpoints of our interval of convergence.

4. Find the interval of convergence for
converges, alternating series test
If x = -1,
diverges, harmonic series
If x = 1,
Interval of convergence [-1, 1)

5. Find the interval of convergence for
If x = -7,
diverges, nth term test
If x = 1,
diverges, nth term test
Interval of convergence (-7, 1)

6. Find the interval of convergence for

7. Find the interval of convergence for
If x = 2,
diverges, harmonic series
If x = 4,
converges, alternating series test
Interval of convergence (2, 4]

8. We can also compare a new series to one we already know.
Harmonic Series -
DIVERGES
p-Series
Converges if p > 1
Diverges if p < 1
Comparison Test for Convergence
Comparison Test for Divergence

9. acts like
for all n
converges by
the comparison test
to the p-series with p = 2.
acts like
diverges by the comparison test to
the p-series with p = 1. (Harmonic)

10. diverges by the comparison test to
haromonic series.
diverges by the nth term test
for divergence.

11. We say…
Converges Conditionally
Converges Absolutely
So…

12. Determine whether the series is converges conditionally,
converges absolutely, or diverges.

13. Determine whether the series is converges conditionally,
converges absolutely, or diverges.
Alternates between -1 and 1.
Diverges.
Converges

14. Determine whether the series is converges conditionally,
converges absolutely, or diverges.
The series diverges by the nth term test for divergence
We could have looked at the absolute value and then the function itself, but we would get the same result.

15. Determine whether the series is converges conditionally,
converges absolutely, or diverges.

16. Determine whether the series is converges conditionally,
converges absolutely, or diverges.
Both alternate…
Since infinite geometric with r = 3/5,
the series converges absolutely.

17. OR Determine whether the series is converges conditionally,
converges absolutely, or diverges. OR

19. We can approximate the value of the sum of an alternating series if…
Error Estimating
We can approximate the value of the sum of an alternating series if…
If we approximate the sum of the first n terms, we will be off by no more than the absolute value of the next term.

20. Find an upper bound for the error if the sum of the first four
terms is used as an approximation to the sum of the series.

21. Find the smallest value of n for which the nth partial sum
approximates the sum of the series within

22. Holy Cow… Lots to remember.
Here’s a chart that might help.
Test
Series
Converge/Diverge
Comments
nth-term
Geometric
Integral
The function f obtained must be continuous, positive, and decreasing.
p-series
Alternating
Applicable only to an alternating series
Useful for series that contain both positive and negative terms.