- Covers HWK 9 - HWK 21
- Handed out Tuesday Nov 30 and due 6 pm Friday Dec 3
- Closed-book but use handout "Frequently Used Maclaurin
Series"
- To be written in one sitting of at most 2 hours
- Topics/terminology/techniques to know:
- What a sequence is and what it means for a sequence to
converge
- Formal definitions of limit statements for sequences
- How to do proofs (of the kind we did) based on the formal
definitions
- What it means for a sequence to be nondecreasing, bounded,
unbounded
- Limit theorems for sequences, including Sandwich Theorem
and Nondecreasing Sequence Theorem
- Limits That Arise Frequently (p 625) and how to verify them
- What a series is and what it means for a series to converge
- Definition of term sequence and partial-sum sequence for a
given series, connection between them and how they relate to
convergence of the series
- What a telescoping series is, how to decide whether one
converges, how to find partial sums for a telescoping series,
and how to find the sum of a convergent telescoping series
- What a geometric series is and how to recognize one, what's
meant by the common ratio of a geometric series, the
convergence criterion for a geometric series, how to find
partial sums for a geometric series, how to find the sum of a
convergent geometric series
- What a p-series is, what a logarithmic p-series is, the
convergence criteria for such series, and why they're valid
- Ways to create new convergent series from old, (sum,
difference, constant multiple), and how to use these operations
in connection with convergence tests
- Convergence tests for series, ideas behind them, and how to
use them: term test, familiar-friend test, tests that apply to
nonnegative series (direct comparison, limit comparison,
integral, ratio, root), ratio test for absolute convergence,
alternating series test. See flowchart on p660.
- Alternating series error estimate and how to use it
- What it means for a series to converge absolutely, to
converge conditionally
- How absolute convergence relates to ordinary convergence
- What a power series is, how to find its radius of
convergence and interval of convergence (including endpoints,
if any), how to differentiate a power series term-by-term (and
where it's legitimate to do so), how to integrate a power
series term-by-term and where it's legitimate to do so, so how
to do algebra and calculus with power series
- How power series differs from a polynomial and how it
differs, in general, from a geometric series
- How to work with the series given in Frequently Used
Maclaurin Series (handout or p696) as power series
- Sections covered
- 8.1. Limits of Sequences of Numbers
includes formal definitions for limit statements involving
sequences
includes Nondecreasing Sequence Theorem
- 8.2.Theorems for Calculating Limits of Sequences
includes Sandwich Theorem;
includes Limits that Arise Frequently p 625
- 8.3. Infinite Series
includes term test, telescoping series, geometric series
- 8.4. Integral Test for Series of Nonnegative Terms
includes p-series, logarithmic p-series
- 8.5. Comparison Tests for Series of Nonnegative Terms
Direct Comparison, Limit Comparison
- 8.6. Ratio and Root Tests for Series of Nonnegative Terms
- 8.7. Alternating Series. Absolute and Conditional
Convergence
see flowchart on p 660 for C-D testing
- 8.8. Power Series
- 8.10 Examples 4, 5 and problems like Problems 1-18, 31-38
only.
- 8.11 Examples 5, 6, 7, 8, 9 and problems like Problems
47-56, 57, 59, 65, 66 only
- Additional Review Problems from Chapter 8 Practice
Exercises p700
- Convergent or Divergent Sequences: 1, 5, 7, 11, 13, 17
- Convergent Series: 19, 21, 23
- Convergent or Divergent Series: 27, 29, 31, 33, 35, 37, 39
- Power Series (part (a) only): 41, 43, 45, 47
- Maclaurin Series: 51, 53, 55, 57, 59, 61, 63
- Indeterminate Forms (part (a) only): 81, 83, 85, 87
- Practice
Test
The practice test should give you a reasonable indication of
roughly what kind of exam and problems to expect. The topics
covered may vary, however, since a test covering this much
material is necessarily selective.
Questions?
Come to my office or
send me e-mail
or post on the FirstClass Q&A conference for the course As
always, to ask about a particular problem from the text, please give
page or section number, give problem number, and make your question
as specific as you can.
Go back to the top
- Alexia Sontag,
Mathematics
- Created by: Kate
Golder
- Wellesley College
- Date Created: June 28, 1999
- Last Modified: November 28, 1999
- Expires: August 31, 2000