1. Does the Series Converge?
10 Tests for Convergence
n-th Term Divergence Test
Geometric Series
Integral Test
p-Series Test
Comparison Test
Limit Comparison Test
Alternating Series Test
Absolute Convergence Test
Ratio Test
Root Test
The test tells you nothing!
Each test has it limitations (i.e. conditions where the test fails).

2. 8.5A Alternating Series – terms alternate in signsOR
NOTE: All an’s are assumed to be positive.

3. Alternating Series - Examples
The alternating harmonic series (will prove to be convergent).
An alternating geometric series (convergent because r = –1/2).
A divergent alternating series (nth-term test).

4. The Alternating Series Test
The series …
Converges if …

5. The Alternating Series Test
Converges if …
“Proof”:

6. Example 1 of the Alternating Series Test
The Alternating Harmonic Series
Decreasing?
Limit?
Therefore, convergent.

7. Example 2 of the Alternating Series Test
Decreasing?
Limit?
Therefore, convergent.

8. Absolute Convergence converges absolutely … converges. if
converges conditionally … if
converges but...
diverges.

9. Absolute Convergence: Example 1
is a convergent alternating series.
Divergent harmonic series, therefore the alternating series is conditionally convergent but not absolutely convergent.

10. Absolute Convergence: Example 2
is a convergent alternating series.
Convergent p-series, therefore the alternating series is absolutely convergent.

11. Absolute Convergence: Example 3
Convergent geometric series, therefore the first series converges absolutely.
Therefore, the original series converges.

12. Absolute Convergence Test: Ex. 1
Convergent geometric series, therefore the first series converges absolutely.
If a series converges absolutely, it is a convergent series.
Note that the first series is NOT an alternating series.

13. 8.5B Approximating Alternating Series
If an alternating series satisfies the conditions of the alternating series test, and SN , the partial sum of the first N terms, is used to approximate the sum, S; then …
The error, RN, is less than the first term omitted.

14. Approximating Alternating Series
Example:
1. Determine the sum of the first 4 terms.

15. Approximating Alternating Series
Example:
2. Estimate the error if 4 terms are used to approximate the sum.
3. Therefore the sum, S, lies between:

16. Approximating Alternating Series
Example:
2. How many terms are needed to make sure the error is less than 0.01?
Therefore, four terms are needed!