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**

** **

** 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]

**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]

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