Review Sheet 1    Math H162

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

Theory that might be on Midterm  I  is marked by asterisks  ***.  Four 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.

Bold face indicates material that will not be on Midterm I, but will be on Midterm II.  This includes the material on power series.

 

Tools

Integration techniques, including integration by parts

            Least Upper Bound axiom for the real number system

            Triangle inequality  [9.1]

            L'Hopital's rule  [25.0]

 

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

Infinite series [2]

Infinite sequence  [2]

Limit of a sequence  [8.1 – 8.3]

Partial sums of a series  [3, 6.1, 7]

Convergent series [6.1, 7]

Properties of sequences [9.2, 16 - 17]

Properties of series  [19, 23.0, 24, 34 – 35, 53, 54.0, 55 - 59]

Divergent series  [6.2, 7]

Absolutely convergent series [51]

Conditionally convergent series [51]

Sum of a series  [3, 6.1, 7, 23.0]

 

Power series  [60 – 61]

Properties of power series [63.0 – 63.1]

 

           

 

Examples

 

         Infinite series  [2, 55 - 59]

 

            Infinite sequences  [2]

 

            Telescoping series  [4]

 

            Power series  [bottom of 5]

 

            Convergent series  [4, 45]

 

            Divergent series  [6.2]

 

            Harmonic series  [10]

 

            Geometric series [11]

 

            p-series [25.1, 31]

 

            power series [62.0]

 

 

 

Theorems--with proofs

***Adding sequences  [9.1 – 9.2]

***Convergence and divergence of geometric series, depending on the ratio [11, 40.8]

***If an increasing infinite sequence has an upper bound, then it converges  [12.1 – 14]

***If a series is absolutely convergent then it is convergent. [52]
 

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

Theorems on subtraction of sequences and multiplication and division of sequences, both by a constant and by another sequence  [9.2]

Comparison test [20, 22, 25.1, 30]

p-series [25.1]

Limit comparison test  [26 - 30]

Integral test  [37.0 – 37.2]

Ratio test  [41 - 42]

Root test  [46 – 47.2]

Alternating series test [49]

 

Calculations

 

            Sum of a finite geometric series

Partial fractions

Telescoping series  [4]


 

Notation

Notation used in writing down limits and series [8.1]