Test II

Study Plan

This is an outline of all the topics and sub-topics, theorems, concepts, and tests, etc. which I'll need to know in order to pass __Test II__ with a good grade.

1. Improper Integrals

   7.8

2. Sequences

   11.1
   1. Limit of Function/Limit of Sequence Relationship Theorem
   2. Infinite Limit of a Sequence
   3. __Limit Laws for Sequences__
   4. Squeeze Theorem for Sequences
   5. Limit of Absolute Value of a Sequence
   6. __Convergence of Sequences__
   7. Upper Bounds of Sequences
   8. Lower Bounds of Sequences
   9. Monotonic Sequences
   10. Bounded Monotonic Sequences Converge

3. Series

   11.2
   1. Definition of Series Convergence
   2. Definition of Series Divergence
   3. Definition of Partial Sums
   1. __Nth Partial Sum__
   2. __Sequence of Partial Sums__
   4. Geometric Series
      1. Common Ration
      2. Convergence of Geometric Series ( $|r| < 1$ )
      3. Formula for the Sum of a Geometric Series
         $\sum_{n=1}^{\infty} ar^{n-1} = \frac{a}{1-r}$
   5. Harmonic Series
   6. __Nth Term Test__
      $\sum_{n=1}^{\infty}a_n$ diverges if $\lim_{n\to\infty}a_n \ne 0$ or _fails to exist._
   7. Adding or Deleting Terms in a Series (doesn't alter convergence/divergence)
   8. Can change __index start value__
   9. __Limit Laws for Series__

4. Integral Test

   11.3
   1. Non-Decreasing Partial Sums
      (only converges if bounded from above)
   2. __Integral Test Theorem__
   3. P-Series
      
      1. Difference between _P-Series & Geometric Series_
   4. __Estimating Sums of Series__
      
      1. Estimating size of remainder
      2. Bounds for remainder in _Integral Test_

5. Comparison Tests

   11.4
   1. __Comparison Test__
   2. __Limit Comparison Test__
   3. Estimating Sums with the Comparison & Limit Comparison Tests

6. Alternating Series

   11.5
   1. Estimating Sums of Alternating Series
      
      1. Alternating Series Estimation Theorem
   2. __Alternating Series Test__

7. Ratio & Root Tests

   11.6
   1. __Absolute Convergence__
   2. __Conditional Convergence__
   3. __Ratio Test__
   4. _Root Test__
   5. __Rearranging Series__

8. Series Testing Strategy

   11.7