Review Sheet 2    Math H162    Wi 05

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

Bold face for material beyond Midterm 1 that could be on Midterm 2.

Theory that might be on Midterm 2 is marked by asterisks  ***.  Two 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.

 

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]

Alternating series  [67]

Absolutely convergent series [70]

Conditionally convergent series [70]

Power series  [76]

Properties of power series []

Radius of convergence of a power series  [82]

Interval of convergence of a power series [85]

Taylor's formula  [104]

Taylor polynomial  [96]

Remainder in Taylor's formula  [98-103]

Taylor series [96]

Polar coordinates [117-118]

           

 

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]

 

            conditionally convergent series [70]

 

         absolutely convergent series [74-75]

Power series  [76-77, 79]

Properties of power series [109.0-116]

Radius of convergence of a power series  [78, 86-88, 113]

Integral of convergence of a power series [86-88]

Taylor's formula  [105-112]

Taylor polynomial  [105-112]

Remainder in Taylor's formula  [98-103, 105-112]

Polar coordinates [117-120]

 

 

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]

***If a series is absolutely convergent then it is convergent. [73-74]
 

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]

Ratio test  [54-56]

Root test  [59-60]

Alternating series test [67-69]

Radius of convergence of a power series  [80-85]

 

Facts

About series and sequences [62-63]

If the power series  a0 + a1x + a2x2 + a3x3 + … has a nonzero radius of convergence  R  and  f(x)  is the function with domain all  x  such that  | x |  < R,  f(x) being defined as the sum of the power series for  x  in that domain, then

         (1)  f(x)  is continuous (at every  x  in its domain)

         (2)  f(x) has a derivative  f'(x), which is the sum of the power series a1 + 2a2x + 3a3x2 +

         (3)  f(x) has an indefinite integral which is given as the sum of the power series  c + a0x +  (a1/2)x2 + (a2/3)x3 + (a3/4)x4 + …

         (4)  The radius of convergence for the power series in (2) and (3) is the same as that of the original power series. [91-93, 113, 115, 116]

***A Taylor series converges to the original function for a given value of x if and only if the remainder term for that value of value of x has limit  0  as n  goes to infinity. [107, 109.0-112]

 

Calculations

 

            Sum of a finite geometric series [26, 64]

Telescoping series  [5, 65]

Convergence tests

            Comparison test [35, 66]

            Limit comparison test [42.1]

            Integral test [52, 53]

            Ratio test [54, 57-59]

         Root test [60-61]

         Alternating series test [70]

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

Radius of convergence of a power series  [89.0-90]

Interval of convergence of a power series [89.0-89.2]

Differentiating and integrating power series  [91]

Taylor polynomials  [94-96, 108-112]

Remainder term  [98, 103, 107, 108-112]

Taylor series  [94-96, 108-116]

Multiplying and dividing power series [113, 115]

Changing between polar and Cartesian coordinates [117-120]

Graphing in polar coordinates [120, 121

Arc length in polar coordinates [122.0-1`23]

Area in polar coordinates  [124-125]
 

Notation

Notation used in writing down limits and series [4]

 

Summation notation using capital sigma  [throughout]