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    1 #DESCRIPTION
2 #Review of Chapter 8 and Polar Coordinates
3 #ENDDESCRIPTION
4
5 #Keywords('Review')
6 DOCUMENT();
8 "PG.pl",
9 "PGbasicmacros.pl",
10 "PGchoicemacros.pl",
12 "PGauxiliaryFunctions.pl"
13 );
14
15 #TEXT(beginproblem());
16
17 TEXT(EV2(<<EOT));
18
19 $BR$BR $BR 20 Here is a short review of numerical series which you may find helpful. 21$BR
22 REVIEW OF NUMERICAL SERIES
23 $BR 24 SEQUENCES$BR
25 A sequence is a list of real numbers.  It is called convergent if it
26 has a limit.  An increasing sequence has a limit when it has an upper bound.
27
28
29 SERIES $BR 30 (Geometric series,rational numbers as decimals, harmonic series,divergence test) 31$BR
32 Given numbers forming a sequence  $$a_1,a_2,...,$$
33 let us define the nth partial sum as sum of the first n of them
34 $$s_n = a_1 +...+ a_n.$$ $BR 35 The SERIES is convergent if the SEQUENCE $$s_1,s_2,s_3,...$$ is. 36 In other words it converges if the partial sums of the series approach a limit.$BR
37 A necessary condition for the convergence of this SERIES is that
38 a's have limit  0.  If this fails, the series diverges. $BR 39 The harmonic series 1+(1/2)+(1/3)+... 40 diverges.$BR This illustrates that the terms $$a_n$$
41 having limit zero does not guarantee the convergence
42 of a series.  $BR 43 A series with positive terms ,i.e. $$a_n > 0$$ for all n, converges$BR exactly when its partial sums have an upper bound. $BR 44 The geometric series $$\displaystyle \sum_{n=1}^{ \infty } r^n$$ 45 converges exactly when $$-1<r<1.$$$BR
46
47 $BR 48 INTEGRAL AND COMPARISON TESTS 49$BR
50 (Integral test,p-series, comparison tests for convergence and divergence, limit comparison test) $BR$BR
51 Integral test:  Suppose $$f(x)$$ is positive and DECREASING for
52 all large enough x.
53 Then the following are equivalent: $BR 54 I. $$\displaystyle \int_1^{ \infty } f(x)dx$$ is finite, i.e. converges.$BR
55 S. $$\displaystyle \sum_{n=1}^{ \infty } \, f(n)$$ is finite, i.e. converges. $BR 56 This gives the p - test: 57 $$\displaystyle \sum_{n=1}^{ \infty } \frac{1}{n^p}$$ converges exactly when 58 $$p > 1 .$$$BR $BR 59 Comparison test: Suppose there is a fixed number K such that$BR for all sufficiently large n:
60  $$0 < a_n < K b_n .$$ $BR 61 62 Convergence. If $$\displaystyle \sum_{n=1}^\infty b_n$$ converges then so does 63 $$\displaystyle \sum_{n=1}^\infty a_n.$$$BR
64
65 Divergence.  If $$\displaystyle \sum_{n=1}^\infty a_n$$diverges then so does
66 $$\displaystyle \sum_{n=1}^\infty b_n$$. $BR 67 68 (Positive series having smaller terms are more likely to converge.)$BR
69
70 Limit comparison test: SUPPOSE:   $$a_n > 0$$, $$b_n > 0$$ and $BR 71 $$\displaystyle \lim_{n \to \infty } \frac{a_n}{b_n} = R$$ exists. Moreover, R is not zero.$BR
72 THEN
73 $$\displaystyle \sum_{n=1}^\infty a_n$$ and $$\displaystyle \sum_{n=1}^\infty b_n$$ $BR 74 both converge or both diverge.$BR
75
76
77 OTHER CONVERGENCE TESTS FOR SERIES $BR 78 (Alternating series test, absolute convergence, RATIO TEST) 79$BR
80 Alternating series test: Suppose the sequence $$a_1,a_2,a_3,...$$ is
81 decreasing and has limit zero.
82 Then $$\displaystyle \sum_{n=1}^\infty {(-1)}^n a_n$$   converges.
83 $BR This applies to (1)-(1/2)+(1/3)-(1/4)+... 84$BR $BR 85 Absolute Convergence Test: IF 86 $$\displaystyle \sum_{n=1}^\infty \vert a_n \vert$$ converges,$BR
87 THEN $$\displaystyle \sum_{n=1}^\infty a_n$$ converges. $BR 88 89 Ratio test:$BR
90 SUPPOSE $$\vert { \frac {a_{n+1}}{a_n}} \vert$$ has limit equal to r. $BR 91 IF $$r < 1$$ then $$\displaystyle \sum_{n=1}^{\infty} a_n$$ CONVERGES.$BR
92 IF $$r > 1$$ the  $$\displaystyle \sum_{n=1}^{\infty} a_n$$ DIVERGES. \$BR
93
94
95 EOT
96
97 &ENDDOCUMENT;
98