The integral test (Sect. 10.3)
Review: Bounded and monotonic sequences.
Application: The harmonic series.
Testing with an integral.
Error estimation in the integral test.
The integral test (Sect. 10.3
Improper integrals (Sect. 8.7)
Review: Improper integrals type I and II.
Examples: I =
1
dx
, and I =
xp
1
0
dx
.
xp
Convergence test: Direct comparison test.
Convergence test: Limit comparison test.
Innite sequences (Sect. 10.1)
Todays Lecture:
Overview: Sequences, series, and calculus.
Denition and geometrical representations.
The limit of a sequence, convergence, divergence.
Properties of seque
Improper integrals (Sect. 8.7)
This class:
Integrals on innite domains (Type I).
dx
The case I =
.
xp
1
Integrands with vertical asymptotes (Type II).
1
dx
The case I =
.
p
0 x
Next class:
Convergence
Innite sequences (Sect. 10.1)
Todays Lecture:
Review: Innite sequences.
The Continuous Function Theorem for sequences.
Using LHpitals rule on sequences.
o
Table of useful limits.
Bounded and monotonic
Innite series (Sect. 10.2)
Series and partial sums.
Geometric series.
The n-term test for a divergent series.
Operations with series.
Adding-deleting terms and re-indexing.
Innite series (Sect. 10.2)
Integrating using tables (Sect. 8.5)
Remarks on:
Using Integration tables.
Reduction formulas.
Computer Algebra Systems.
Non-elementary integrals.
Limits using LHpitals Rule (Sect. 7.5).
o
Integrating
Integration by parts (Sect. 8.1)
Integral form of the product rule.
Exponential and logarithms.
Trigonometric functions.
Denite integrals.
Substitution and integration by parts.
Integral form of the p
Review for Exam 3.
5 or 6 problems.
No multiple choice questions.
No notes, no books, no calculators.
Problems similar to homeworks.
Exam covers: 8.3, 8.4, 7.5, 8.7, 10.1.
Trigonometric substitutions
Trigonometric substitutions (Sect. 8.3)
Substitutions to cancel the square root
Integrals involving a2 x 2 : Use x = a sin().
Integrals involving a2 + x 2 : Use x = a tan().
Integrals involving x 2 a2
Trigonometric integrals (Sect. 8.2)
Product of sines and cosines.
Eliminating square roots.
Integrals of tangents and secants.
Products of sines and cosines.
Product of sines and cosines
Remark: There
Review for Exam 2.
5 or 6 problems.
No multiple choice questions.
No notes, no books, no calculators.
Problems similar to homeworks.
Exam covers: 7.4, 7.6, 7.7, 8-IT, 8.1, 8.2.
Solving dierential equa