Lecture: Telescoping series, careful definition of
limit of a sequence being ,
etc. Difference and relation between infinite Sequences and
Series -- repeating this message. The Harmonic Series.

Lecture/Discussion: What is wrong with Guido Ubaldo's
"Proof" that 0=1:

0

=

0 + 0 + 0 + 0 + ...

=

(1-1)+(1-1)+(1-1)+...

=

1+(-1+1)+(-1+1)+...

=

1 + (0) + (0) + (0) + (0) + ... = 1

A discussion often follows; ask students to decide,
discuss (a friendly debate often ensues): which
equal sign(s) is the source of the problem? Why?

Lecture: Test for Divergence, and the difference
between the converse and the contrapositive. (A different which "It
cannot be emphasized too often to instructros, that it cannot be
emphasized too often to the students...")

Week 8:

Days 1 and 2:

Converting infinite decimals (e.g. 1.21888...)to fractions.
Final discussion of The Race.

State as Fact: p-series Test. ("Be able to
use it"; improper integrals covered later).

Lecture: Comparison Test (CT)and Limit Comparison Test (LCT).

Examples: E.g. writing (ln(n))/(n^{3}) as..? as [(ln(n))/(n)] times [1/(n^{2})]

What about (ln(n))/((n^{2})? Not
so simple? Notice log is "beaten out" by even root powers.

What about
(ln(n))/((n^{1.1})?
what about (ln(n))/((n^{p})?

Days 3 and 4:

Examples with CT and LCT.
Series of 1/((2^{n})?
Series of 1/((2^{p} + 1)? (two ways).

Examples like series of (3n^{2} +
2n) / [square-root(n^{7} + 5)].

Rule of Thumb: "Look at top power in
numerator, and denominator" -- then check rigorously with LCT.

Project #2 is due around this time, depending on the semester.

Week 9:

Midterm is held this week -- Midterm includes
Activity-type problem(s) (30-35%)

Project #2 is due.

Solutions to Medical Dosage, Stack Up,
and Nonbook Problems.

Review for Midterm.

Lecture: Finite Absolute Convergence and Ratio &
Root Tests; Cover "Strategy".

Week 10:

Activity: Students work in groups to solve the
Midterm's Activity-type problem and more advanced add-ons. Typically
this goes very well, very appreciated by the students.

Project #3 is assigned: Pi in
the Sky (due Week 13).

[Sample Grading
Scheme for Pi in the Sky for instructors/graders].

[under construction: Clarifications about languate of Pi in
Sky project][ALSO: student evals for Pi et al]

Lecture: Improper Integrals; The Integral Test; Power
Series.

Week 11:

Lecture: Taylor Series; Taylor Series Approximations