1 / 19

11.3

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

marcel
Download Presentation

11.3

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


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

  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. • is divergent since • Example: The series • However, • does not imply convergence!

  3. 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.

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

  5. 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:

  6. 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.

  7. 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.

  8. 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.

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

  10. 11.7 • Strategy for Testing Series

  11. 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?

  12. Summaryof all tests:

  13. 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.

  14. More examples

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

  16. Example 2 Since an is an algebraic function of, we compare the given series with a p-series. The comparison series for the Limit Comparison Test is where

  17. Example 4 Since the series is alternating, we use the Alternating Series Test.

  18. Example 5 • Since the series involves k!, we use the Ratio Test.

  19. Example 6 Since the series is closely related to the geometric series , we use the Comparison Test.

More Related