Test III Review Sheet for
Colorado State University, Pueblo
Math 224 — Calculus II — Fall 2014

Definitions/Theorems/Ideas:

  1. sequences: recurrence relation, explicit formula
  2. limit of a sequence: both general idea (convergent and divergent) and formal definition
  3. geometric sequence
  4. limit laws for sequences
  5. squeeze theorem for sequences
  6. bounded monotonic sequences and their convergence
  7. series
  8. limit of s series: idea in terms of limit of the sequence of partial sums; convergent and divergent, again.
  9. geometric series and when it converges and to what
  10. telescoping series
  11. tests about convergence of series:
    1. the divergence test
    2. the integral test
    3. the ratio test
    4. the root test
    5. the comparison test
    6. the limit comparison test
    7. the alternating series test
    Always check all their hypotheses
  12. alternating series
  13. the harmonic and alternating harmonic series
  14. absolute and conditional convergence of series
  15. approximating functions with polynomials: linear, quadratic, and higher-order approximations
  16. then remainder of a Taylor polynomial
  17. a power series [centered at $x=a$]
  18. Taylor's Theorem
  19. estimating error in power series
  20. the theorem about convergence of a power series: Interval and Radius of convergence (IoC and RoC)
  21. the theorem(s) about operations on power series: CMR, addition, multiplication, composition, division, differentiation, integration, etc.
  22. the Taylor series and Maclaurin series of a function
  23. binomial series and the binomial coefficient
  24. more specific applications of power series:
    1. limits
    2. representing real numbers as series
    3. solving differential equations
    4. representing functions as power series and, conversely, finding the function represented by a power series
You must be able to define, work with, recognize, and apply all of the above.

Specific things not to forget:

  1. Always check all the hypotheses of a test you use.
  2. Know how to prove from the formal definition the value of a sequence limit.
  3. Usually we have to check the endpoints of an IoC by hand.
  4. Keep an eye on the center of a power series: put the terms $(x-a)^k$ in the formula; use a good center when using a power series to approximate something.
  5. How to use the Squeeze Theorem.
  6. How to use all the tests for convergence of a series.
  7. How to compute the RoC and IoC of a given power series.
  8. How to compute the power series of a function and the function represented by a power series.
  9. How to use power series in applications, such as:
    1. approximating functions or specific real numbers
    2. solving differential equations
    3. finding limits

Memorize:

  1. the power series for $e^x$, $\sin(x)$, $\cos(x)$, and the geometric series (so the function is $\frac{1}{1-x}$) [I will put on the test any other complicated power series you might need beyond those four.]
  2. the "absolute convergence implies regular convergence" theorem
  3. things about $p$-series
  4. the harmonic series and alternating version
  5. the divergence test

The Test Will Have:

  1. A summary of the table on page 673 of the book (without the lines on geometric series, the divergence test, $p$-series, and absolute convergence). So you don't need to memorize that table.
  2. There will definitely be problems along the lines of: