Final Review – Exam 3 Sequences & Series Improper Integrals

Chapter 10: Sequences and Series 1)Sequences 2)Series 3)Sequence of Partial Sums 4)Named series: geometric, telescoping, p-series. 5)Tests for testing series 6)Absolute and Conditional Convergences 7)Error bound estimations for integral test and alternating series test. Please refer to lecture slides (notes) or textbook for details.

Sequences (10.1 & 10.2) Convergence of sequences Definitions of bounded and monotonic sequences Theorem: Every bounded and monotonic sequence is convergent.

Example 1 (exam 3 problem 2) a) monotonic and convergent b) monotonic and divergent c) bounded and convergent d) bounded and divergent e) monotonic, bounded and convergent f) monotonic, bounded and divergent

Sequence of Partial Sums and Series (10.3) Infinite series: Partial Sums:

Sequence of Partial Sums and Series (10.3) For the infinite series and write

Example 2 (exam 3 problem 1) F F B A.Diverges B.Converges to 0 C.Converges to 1 D.Converges to 2 E.Converges to 3 F.Converges to 6 G.Converges to 12

Named Series Geometric series Telescoping series

Example 3 (exam 3 problem 7) Determine whether the series is convergent or divergent. If it’s convergent, find its sum if possible.

Named Series

Example 4 (exam 3 problems 5 and 6) Determine whether the series is convergent or divergent.

Absolute & Conditional Convergence (10.6) Remark: Any positive term series that is convergent is absolutely convergent. A positive term series cannot be conditionally convergent.

How to handle the questions ? Determine whether the series is Conditionally Convergent, Absolutely Convergent or Divergent. 1)For absolute convergence:  Use Ratio Test or Root Test, or  Use the definition. 2)For conditional convergence:  Use the definition. Determine whether the series is Convergent or Divergent:  use any test that applies.

Example 5 (exam 3 problem 8) Prove that the series is conditionally convergent.

Example 6 (exam 3 problem 9) Show that the series is absolutely convergent.

Estimation for Integral Test (10.4) The exact value of the sum is bounded as follows:

Alternating Series Estimation Theorem (10.6) the first neglected term In other words, the remainder (the error) is the less than or equal to the first neglected term. It is the first term that is not used in the approximation.

Example 7 (exam 3 problem 3)

Type I - Improper Integrals (8.8) If the limits exist we say the integral converges or is convergent. Otherwise, we say the integral diverges or is divergent.

Type II - Improper Integrals (8.8) (provide that the limits on the right side exist) (provide that the integrals on the right side converge)

Example 8 (exam 3 problem 10) Evaluate the integral.