Exam 3 will be held during the lab session on Wednesday November 29 in Room 143 Altgeld.

Remark on Limit Calculations:

I will not be super rigid in appraising the way in which you evaluate limits. What will be required is a good explanation of how you are determining what a limit might be.

I will not accept abuses of notation. Thus infinity/infinity will be marked wrong. Similarly F(infinity)-F(1) will be marked wrong for the integral of f from 1 to infinity if F is an antiderivative of f.

The following topics will be covered on the exam:

Chapter 9

• Section 9.1:
  • The main technique in this section is the computation of the nth degree Taylor Polynomial P_n of a function f about a base point x₀. Make sure you know how to compute P_n.
  • Know the sense in which P_n approximates f: both f and P_n and their derivatives up to order n agree at the base point x₀. This is called "contact of order n at x₀".
  • Accuracy of using P_n in place of f near x₀ should increase as n is increased.
• Section 9.2:
  • Theorem 2 (Taylor's Theorem) is the fundamental result in this section. Know it forwards and backwards.
• Be able to use theorem 2:
  • Know how to find K_n+1
  • Know how to estimate error on a fixed interval of x's
  • Know how to find an interval that produces an error smaller than a given number.

Chapter 10

• Section 10.1:
  • What makes an integral improper?
  • How do we define the value of an improper integral and what do convergence and divergence mean? Know how to impliment these concepts in the case of specific improper integrals, i.e. testing convergence or divergence, and computing the integral if it is convergence.
  • What do convergence and divergence mean if an integral is improper is more than one way?
• Section 10.2:
  • Know the comparison theorem and how to use it in assessing convergence and divergence.
  • What is the p-test for improper integrals?
  • Know Theorem 2 on absolute convergence of integrals
  • Know how to estimate the size of the "tail" of an improper integral, i.e. for what value of a is the improper integral from a to infinity less than a specified error?

Chapter 11

• Section 11.1:
  • What does one mean by the convergence of a sequence?
  • Know Theorems 1 and 2 and how to use them in practical calculations
  • Be able to compute the limit of sequences, possibly using L'Hopital's rule.
• Section 11.2:
  • What does one mean by the convergence and divergence and the sum of a series?
  • What is a harmonic series?
  • Know how to identify and compute the sum of a geometric series
  • Know Theorem 5 for breaking complex series into sums of simpler series
  • Know Theorem 6 - the nth term test for non-convergence
• Section 11.3: Series of Positive Terms
  • Know the comparison test for series and how to use it
  • Know Theorem 8 on the integral test and the estimates it gives
  • What is the p-test for series?
  • Know how to use the ratio test to test for convergence.
• Section 11.4: General Series and Alternating Series
  • What is meant by absolute and conditional convergence. Be able to test for these is simple problems
  • Be sure to know Theorem 11 on Alternating Series, including the error estimate. You should be able to use this in simple cases to estimate the error between the sum of an alternating series and the nth partial sum
• Section 11.5: Power Series
  • What is a power series, i.e. base point, coefficients?
  • What does one mean by the radius of convergence of a power series?
  • Know how to comput the radius of convergence of a power series using the ratio test.