Part 1

• Bressoud: A Radical Approach to Real Analysis, chapters 3-4
  This is the primary text for this unit.
• Sprecher: Elements of Real Analysis, chapters 6-7 & chapter 4
  This is a secondary text for this unit.

• Differentiability and Continuity
• The Newton-Raphson method
• Definitions: differentiable, derivative
• The negation of differentiability
• Theorems on derivatives of sums, differences, products, quotients
• The intermediate value property (IVP)
• Cauchy's generalized mean value theorem
• L'Hospital's rules
• The Lagrange and Cauchy forms of Taylor's Theorem
• Definitions: Continuity at a point, on an interval, on a set (Use Sprecher's definition 24.1 when Bressoud's definitions do not apply.)
• The Intermediate Value Theorem (IVT)
• Piecewise monotonic function
• Continuity of sums, differences, products, quotients, and compositions of continuous functions
• Theorem: Differentiability implies continuity.
• The Extreme Value Theorem
• Fermat's theorem on extrema
• Rolle's theorem
• The Mean Value Theorem (MVT)
• Theorem: If f' exists on [a,b], then f' has the IVP.
• The converse of the IVT for piecewise monotonic functions
• Sequences and Series
  • Two meanings of a₁+a₂+a₃+ ...
  • The limit of the partial sums (if it exists) may be called the "sum" or the "value" or the "limit" of the series.
  • Convergence is a "tail event."
  • Two limits that equal e (Sprecher p. 91)
  • Absolute convergence and conditional convergence (*)
  • The basic tests for convergence of a series
    • The "n^th-term test"
    • The Cauchy criterion
    • Leibniz's alternating series test
    • The comparison test
    • The ratio test
    • The limit ratio test
    • The root test
    • The limit root test
  • Series of functions
    • Power series
    • Trigonometric series
  • The set of convergence of a power series
  • Formula for the radius of convergence
  • Upper and lower limits: lim sup and lim inf
  • Hypergeometric series and hypergeometric power series
  • The set of convergence of a hypergeometric power series can be completely determined by the theorems in sect 4.3 of Bressoud, including...
  • Gauss's test
  • The finer tests for convergence of series
    • Boundedness test for series of nonnegative terms
    • Cauchy's condensation test
    • The integral test
    • Dirichlet's test (proof uses Abel's lemma)

Part 2

Pick up your copy of the exam outside my office, Olin 307, when you are ready to take it, and sign it out on the signout sheet. You will have 2-3 hours to take it. Return it to my office ASAP after you finish it, and note the return time on the signout sheet.

In addition to problems whose solution may use the mathematics listed above, you may find it useful to know the following:

• Summation by parts
• Finite geometric sums
• Telescoping sums
• Errors and error bounds for series:
  • Lagrange form of the remainder for Taylor series
  • Cauchy form of the remainder for Taylor series
  • An improper integral bound for the error when the integral test implies convergence
  • Successive partial sums of an alternating series of terms whose magnitudes decrease to zero surround the sum (limit) of the series.

Click here to get a postscript or pdf view of the instruction page: (ps) (pdf) (*)