1. Convergence or Divergence of Infinite Series

2. Definition of Convergence for an infinite series:
Let be an infinite series of positive terms.
The series converges if and only if the sequence of partial sums,
, converges. This means:
Divergence Test:
If , the series diverges.
Example: The series
is divergent since
However,
does not imply convergence!

3. Special Series

4. Geometric Series: The Geometric Series: converges for
If the series converges, the sum of the series is:
Example: The series with and converges .
The sum of the series is 35.

5. p-Series: The Series: (called a p-series) converges for
and diverges for
Example: The series is convergent. The series is divergent.

6. Convergence Tests

7. 11.3 11.4 11.5 11.6 The Integral Test The Comparison Tests
Alternating Series
11.6
Ratio and Root Tests

8. Integral test:
If f is a continuous, positive, decreasing function on with
then the series converges if and only if the improper integral converges.
Example: Try the series: Note: in general for a series of the form:

9. Comparison test:
If the series and are two series with positive terms, then:
If is convergent and for all n, then converges.
If is divergent and for all n, then diverges.
smaller than convergent is convergent
bigger than divergent is divergent
Examples: which is a divergent harmonic series. Since the original series is larger by comparison, it is divergent.
which is a convergent p-series. Since the
original series is smaller by comparison, it is
convergent.
Makes sense!

10. Limit Comparison test:
If the the series and are two series with positive terms, and if
where then either both series converge or both series diverge.
Useful trick: To obtain a series for comparison, omit lower order terms in the numerator and the denominator and then simplify.
Examples: For the series compare to which is a convergent p-series.
For the series compare to which is a divergent geometric series.

11. Alternating Series test:
If the alternating series satisfies:
and then the series converges.
Definition: Absolute convergence means that the series converges without alternating (all signs and terms are positive).
Example: The series is convergent but not absolutely convergent.
Alternating p-series converges for p > 0.  
Example: The series and the Alternating Harmonic series are convergent.

12. Ratio test: If then the series converges; If the series diverges.
Otherwise, you must use a different test for convergence.
If the limit is 1, the test is inconclusive and a different test is required.
Specifically, the Ratio Test does not work for p-series.
Example:

13. Strategy for Testing Series
11.7
Strategy for Testing Series

14. Summary: Apply the following steps when testing for convergence:
Does the nth term approach zero as n approaches infinity? If not, the Divergence Test implies the series diverges.
Is the series one of the special types - geometric, telescoping, p-series, alternating series?
Can the integral test, ratio test, or root test be applied?
Can the series be compared in a useful way to one of the special types?

15. Summary of all tests:

16. Example
Determine whether the series converges or diverges using the Integral Test.
Solution: Integral Test:
Since this improper integral is divergent, the series  (ln n)/n is also divergent by the Integral Test.

17. More examples

18. Example 1
Since as an  ≠ as n  , we should use the Test for Divergence.

19. Example 2: Using the limit comparison test
Since an is an algebraic function of n, we compare the given series with a p-series.
The comparison series for the Limit Comparison Test is
where

20. Example 3
Since the series is alternating, we use the Alternating Series Test.

21. Example 4
Since the series involves k!, we use the Ratio Test.

22. Example 5 Since the series is closely related to the geometric series
, we use the Comparison Test.