Fall 2009, Math 104

November and December, 2009

Be sure to check back often, because assignments may change!

All section and page numbers refer to sections from Calculus: Early Transcendental Functions, Smith and Minton, 3rd Edition.

Due Monday 11/2 at 9am

Section 8.2: Infinite Series

• To read: Finish the section, then re-read the whole thing-- really work to make sense of it all.

E-mail Subject Line: Math 104 Your Name 11/2

What does the kth Term Test tell you about each of the following series? Explain.

1. sum(sin(k), k=0..infinity)
2. sum(1/k , k=1..infinity)

Reminders:
• Monday is the deadline to receive 90% on the antidifferentiation exam. If you haven't passed it yet, don't let this opportunity go by without the attempt -- several, if necessary!

Due Wednesday 11/4 at 9am

Section 8.3: The Integral Test and Comparison Tests

• To read: Through Example 3.4.
• Be sure to understand: How you can use the integral test to determine the convergence or divergence of a sum or an itnegral; (for Thursday) how it's used to approximate the value of a series.

E-mail Subject Line: Math 104 Your Name 11/4

1. Explain in a couple of sentences (in your own words, of course) why the integral test makes sense. That is, explain the idea behind the integral test.

Reminders:
• Bring questions on PS 9.
• Take advantage of tutoring hours

Due Friday 11/6 at 9am

Section 8.3: The Integral Test and Comparison Tests

• To read: Through Example 3.7
• Be sure to understand: How to use the comparison test to determine convergence

E-mail Subject Line: Math 104 Your Name 11/6

1. Explain in a couple of sentences why the Comparison Test makes sense. That is, explain the idea behind it.

Reminders:

• Begin PS 10. This is a group assignment - remember the usual guidelines: variety in group make-up; 2-3 people; don't divvy; switch and label primary authors.

Due Monday 11/9 at 9am

Section 8.3: The Integral Test and Comparison Tests

• To read: While I will not be covering the Limit Comparison Test in class, do read the rest of the section -- while it is not obvious how it can be used in approximating a series, it does resolve some of the difficulties we can run into with the Comparison Test. Also, re-read the entire section, making sure you understand each example and looking for more nuance than you saw on the first read-through.

Due Wednesday 11/11 at 9am

Section 8.3: The Integral Test and Comparison Tests

• To read: In class, I covered using the Integral and Comparison Tests to help approximate convergent series in more detail than the text did. Read through your notes and in-class work related to this.

E-mail Subject Line:Math 104 Your Name 11/11

Consider the series sum( 12+3j, j=0..infinity). (This is Maple notation for a sum or series).

1. Show that the series converges.
2. Estimate the limit of the series within 0.01.
3. Is your estimate an over- or under- estimate?

Reminders:

• Bring questions on PS 10

Due Friday 11/13 at 9am

Continue Working on Project 1

Reminders:

• Begin PS 11

Due Monday 11/16 at 9am

Section 8.3: The Integral Test and Comparison Tests

• To read: Again, re-read through your notes and in-class work related to the choosing which series to use and also to finding upper and lower bounds for the limit of a convergent series, and to estimating a convergent series within a certain margin of error (i.e. within a certain error bound).

E-mail Subject Line: Math 104 Your Name 11/16

Consider the series sum( 1(k4+5), k=1..infinity). (This is Maple notation for a sum or series).

1. Show that the series converges.
2. Estimate the limit of the series within 0.01.
3. Is your estimate an over- or under- estimate?

Reminder:

• Just a reminder of the obvious: each of you should be doing your best to contribute equally on the project.

Due Wednesday 11/18 at 9am

Section 8.4: Alternating Series

• Be sure to understand: The statement of the Alternating Series test

E-mail Subject Line: Math 104 Your Name 11/18

1. Does sum( (-1)ksqrt(k), k=1..∞) converge or diverge?
2. How closely does S100 approximate the series sum((-1)k (1/k), k=1 .. infinity) ? Why?

Reminder:

• You should have the mathematics behind project 2 solved by Wednesday morning, so you can begin writing your response. I once again encourage you to bring me a draft.

Due Friday 11/20 at 9am

Section 8.5: Absolute Convergence and the Ratio Test

• To read: Through Example 5.3.

E-mail Subject Line: Math 104 Your Name 11/20

1. Give an example of a series that is conditionally convergent. Explain.
2. Give an example of a series that is absolutely convergent. Explain.
Reminders:
• Begin PS 12
• If you still have the antidifferentiation exam hanging over you, Friday is the last day you can receive 75% on it. After that, it goes down to 50% until the end of classes.

Due Monday 11/23 at 9am

Section 8.5: Absolute Convergence and the Ratio Test

• To read: The section The Ratio Test and the section Summary of Convergence Tests. Optional: the section The Root Test.
• Be sure to understand: How to use the ratio test to determine convergence and divergence, and when the ratio test is inconclusive.

E-mail Subject Line: Math 104 Your Name 11/23

1. Explain in a couple of sentences why the Ratio Test makes sense.
Reminders:
• Remember to attach a copy of the checklist to the front of your project with a paperclip.
• Just a reminder that Exam 3 is the Thursday after Thanksgiving break. While I know you just now finishing Project 2, you may want to look at your post-break schedule and decide whether you want to get a good start on preparing yourself.

Due Wednesday 11/25 at 9am

Thanksgiving Break!

Due Friday 11/27 at 9am

Thanksgiving Break!

Due Monday 11/30 at 9am

Section 8.6: Power Series

• To read: Through Theorem 6.1

E-mail Subject Line: Math 104 Your Name 11/30

1. How do power series differ from the series we have looked at up to this point?
2. What is the interval of convergence of a power series? Why have we not discussed the interval of convergence before? Explain in your own words.
Reminders:
• Begin studying for the Exam, if you haven't already.

Due Wednesday 12/2 at 9am

Bring Questions for Exam 3

Reminders:

• PS 12 will not be collected, but it will be covered on the exam.
• As always, you may have handwritten notes on one side of a standard sheet of paper, and may begin the exam at 12:30.
• Get as many questions resolved before class on Wednesday as possible, through office hours and tutoring hours.

Due Friday 12/4 at 9am

Section 8.6: Power Series

• To read: Through Example 6.4

E-mail Subject Line: Math 104 Your Name 12/4

1. Suppose the power series sum(bk(x-c)k, k=0..∞) has a non-trivial and non-infinite interval of convergence. Where will the midpoint of that interval be?
Reminder:
• Begin PS 13

Due Monday 12/7 at 9am

Section 8.6: Power Series

• To read: Finish the section

E-mail Subject Line: Math 104 Your Name 12/7

1. As you have read, we can differentiate and integrate a convergent power series term-by-term. Why is this not obvious?
2. Why might you want or need to use a power series?

Due Wednesday 12/9 at 9am

Section 8.7: Taylor Series

• Be sure to understand: The definition of a Taylor series

E-mail Subject Line: Math 104 Your Name 12/9

1. How does a Taylor series differ from a Taylor polynomial?
2. Give two good reasons for writing a known function ( such as cos(x) ) as a power series.

Reminder:

• Bring questions on PS 13 to class

Due Friday 12/11 at 9am

Section 8.8: Applications of Taylor Series

E-mail Subject Line: Math 104 Your Name 12/11

1. If you wanted to find exp(sqrt(3)) using a Taylor series, what would you as a base point?
Reminder:

Here ends the reading for the Semester!

Janice Sklensky
Wheaton College
Department of Mathematics and Computer Science
Science Center, Room 101A
Norton, Massachusetts 02766-0930
TEL (508) 286-3973
FAX (508) 285-8278
jsklensk@wheatonma.edu

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