Math 142

Chapter 10 Homework

Homework will be posted as it becomes available.

section

remark

homework problems
10.1

sequences

 Review indeterminate forms (L'Hopital's Rule) from Calculus I:
Online HW
need more practice? problems (suggestion: the circled ones) and solns
Varberg, Purcell, Ridgon (8th ed) § 10.1
10.2

Numerical
Infinite
Series
Basics

----

Geometric Series
Telescoping Series

  
  • To find the sum (or partial sum) of a geometric series ∑ a rn, use the method from class of finding a formula (without ...-sign or ∑-sign) for sn by creating cancellation heaven with sn- r sn. Do not use some need-to-memorize formula from some book.
  • For this homework set, unless otherwise specified, when we say the nth partial sum sn, we mean the sum of the first n terms. Below are some examples to clarify.
    1. If the sum starts with a1, then sn = a1 + a2 + a3 +... + an.
    2. If the sum starts with a0, then sn = a0 + a1 + a2 +... + an-1.
    3. If the sum starts with a2, then sn = a2 + a3 + a4 +... + an+1.
    4. If the sum starts with a17,
      then sn = a16+1 + a16+2 + a16+3 +... + a16+n.
Online HW
need more practice? problems and solutions
Covers: geo. series, telescoping series, and nth term test for divg.
Varberg, Purcell, Ridgon (8th ed) § 10.2.
Postive-termed Numermical Series
10.3

Integral Test

 Online HW
need more practice? problems and solutions
Varberg, Purcell, Ridgon (8th ed) § 10.3
10.2&10.3 A practice quiz. (soln)
10.4

Direct Comparison Test (DCT)

and

Limit Comparison Test (LCT)

for Series

 Online HW
Off-Line HW

 

helpful intuition needed:

10.3vs10.4

 Do problems 40 and 41 of these DCT/LCT for series practice problems (soln), which are from another textbook.
  1. First do 40a and 41a.
  2. Next do 40b and 41b, using the DCT.
    To pick a series ∑ bn for comparsion, recall (from class) the helpful intuition about growth rates of the log&exp functions.
  3. Now do 40b and 41b, using the LCT.
    To pick a series ∑ bn for comparsion with the LCT way, see your choices for ∑ bn using the DCT way.
Stewart (6th ed, ET), §11.4.
need more practice? problems and solutions
Do at least: 1,2,3, 5, 7, 9, 11, 13, 17, 21, 29, 31, 39, 43
Stewart (6th ed, ET), §11.4.
A practice quiz. (soln)
Arbitrary-termed Numerical Series

Thus far we looked at positive-termed series, which were either convergent (to a finite number) or divergent (to infinity). Now we will look at arbitrary-termed (not necessarily positive-termed) series, which can be either:
absolutely convergent (AC), conditionally convergent (CC), or divergent (DVG).
10.5

Ratio/Root Tests

Online HW
A practice quiz. (soln)
10.4&10.5

DCT/LCT&
Ratio/Root

need more practice? problems and solutions
Varberg, Purcell, Ridgon (8th ed) § 10.4
A practice quiz. (soln)
The first problem can be (correctly) done using either DCT or LCT.
Which way do you like better?
10.6

Alternating Series Test

 Online HW
need more practice? problems and solutions
Varberg, Purcell, Ridgon (8th ed) § 10.5
Review of Numerical Series: § 10.2-10.6
10.2-10.6:

Review Problems
for
Exam Preparation

 Determine whether the given series is AC, CC, or divergent.

From 38 Serious Series Problems, do at least the odd's (omit 29).
Give the problems a good honest shot before peeking at the solns.
Stewart (6th ed, ET), §11.7.

Determine whether the given series is AC, CC, or divergent.

Do each of these 15 Serious Series Problems.
Give them a good honest shot before peeking at hints.  [pdf files merged]

Exam 2 is over § 10.1-10.6
10.7

Power Series

 Online HW
Practice Problem 0 for §10.7: Power Series (soln)
Problems 0.1 - 0.3 are for Part 1 while 0.4 is for Part 2.
Part 1:

basics (AC/CC/DVG)

 To start, 7 (=1+6) homework problems.
Sol'n to the first problem as well as the other 6 problems.
Part 2:

Operations on Power Series

 From another book §11.9 homework set do: 1, 5, 7, 11, 13, 14, 15, 23.
If asked to find a power series representation of a function and a center is not specified, choose whatever center you want (the solns chose zero). (soln)
Stewart (6th ed, ET), §11.9.
10.TS

Taylor Series

(in book: 10.8-10.10)

Needed
Resources

and

Solutions
  • Practice Problem 0 for Taylor Series.   [soln]
    • a-c is for Taylor Polynomials
    • d-e is for Taylor Series without Remainder
    • f is for Taylor Series with Remainder
  • Taylor Series Homework (3 problems, each with parts a-i)
    • a-b is for Taylor Polynomials
    • c-f is for Taylor Series without Remainder
    • g-i is for Taylor Series with Remainder
    Give the problems an honest shot before peaking at solutions for function: (1),  (2),  (3).
  • A nice read on Taylor Polynomials (p50-54) and Taylor Series (p54-61) from How to Ace the rest of Calculus
    by Adams, Hass, and Thompson.
  • Assorted online homework labeled
    10.TS (for Chapter 10, Taylor Series).
Part 0:

Intro. to
Taylor Polynomials

Review the Introduction to Taylor Polynominals.

Nice videos visually illustrating convergence for the
Taylor Series for Sine and the Geometric Taylor Series.

Part 1:

Taylor Polynomial

Online HW: 10.TS Part 1: Taylor Polynomials.

Do parts a-c from the Practice Problem 0 for Taylor Series.

Do parts a-b for each of the 3 problems in Taylor Series HMWK.

Part 2:

Taylor Series
(Without Remainder)

Online HW: 10.TS Part 2.

Do parts d-e from the Practice Problem 0 for Taylor Series.

Do parts c-f for each of the 3 problems in Taylor Series HMWK.

Part 3:

Taylor Remainder

Online HW: 10.TS Part 3.

Do part f from the Practice Problem 0 for Taylor Series.

Do parts g-i for each of the 3 problems in Taylor Series HMWK.

Part 4:

Commonly Used Series

Online HW: 10.TS Part 4.

No off-line homework.

Solutions to a sample Taylor Remainder Quiz (Fall 2016).

Findable from URL: http://people.math.sc.edu/girardi/