Review Sheet 1    Math H162    Wi 05

[n] means the topic is found on acetate n.

Theory that might be on Midterm  I  is marked by asterisks  ***.  Six items are marked.  One of them will be on the exam.

To the exam you may bring one 8-1/2 x 11 sheet of paper on which are written the various convergence and divergence teats.

The ratio test and root test will not be on midterm I.

 

Tools

Integration techniques, including integration by parts

            ***Least Upper Bound axiom for the real number system [30]

            L'Hopital's rule

Partial fractions

            Induction [13-18]

 

Definitions (know them and be able to write them correctly)

Infinite series [2, 3, 4]

Infinite sequence  [2]

***Limit of a sequence  [7, 9, 10]

Terms of a series  [3, 4]

Partial sums of a series  [3, 4]

Convergent sequence [23]

Divergent sequence  [23]

Convergent series [23]

Divergent series  [23]

Nondecreasing sequence  [29]

Sum of a series  [4,5, 23]

           

 

Examples

 

         Infinite series  [4, 24, 34, 40, 49, 50]

 

            Infinite sequences  [2, 3]

 

            Limits of sequences  [8, 11]

 

            Telescoping series  [5]

 

           

            Divergent series  [38, 39]

 

            Harmonic series  [38]

 

            Geometric series [24-28]

 

                        First term  [24]

                        Ratio  [[24]

                        Sum  [24]

 

            p-series [41]

 

           

 

 

Theorems--with proofs

***Convergence and divergence of geometric series, depending on the ratio [24-28]

***If an nondecreasing infinite sequence converges if and only if it is bounded  [29-32]

***A series of positive terms converges if and only if its sequence of partial sums in bounded [32-33]

***Test for convergence of p-series [49]
 

Theorems--no proofs (know what they mean and be able to state and use them correctly)

Theorems on addition and subtraction of sequences and series and multiplication by a constant [39]

If a series of positive terms is rearranged (put into a different order) then the original series and the rearranged series either both converge (and to the same sum) or they both diverge  [45-47]

If a series converges, then the individual terms of the series have limit zero. [sorry, I can't find this]

Comparison test [35-36]

Limit comparison test  [42.1]

Integral test  [48, 51.0-51.2]

 

Calculations

 

            Sum of a finite geometric series [26]

Telescoping series  [5]

Convergence tests

            Comparison [35]

            Limit comparison  [42.1]

            Integral test [52, 53]

Using highest powers in numerator and denominator  [43-44]


 

Notation

Notation used in writing down limits and series [4]