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**

** **

** I**nfinite 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 a**_{0}** + a**_{1}**x
+ a**_{2}**x**^{2}**
+ a**_{3}**x**^{3}**
+ … 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 a**_{1}** + 2a**_{2}**x
+ 3a**_{3}**x**^{2}**
+**

** (3) f(x) has an indefinite integral which
is given as the sum of the power series
c + a**_{0}**x + (a**_{1}**/2)x**^{2}**
+ (a**_{2}**/3)x**^{3}**
+ (a**_{3}**/4)x**^{4}**
+ …**

** (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]**