View of /trunk/NationalProblemLibrary/Rochester/setSeries6CompTests/review.pg

    1 #DESCRIPTION
    2 #Review of Chapter 8 and Polar Coordinates
    3 #ENDDESCRIPTION
    4 
    5 #Keywords('Review')
    6 DOCUMENT();
    7 loadMacros(
    8 "PG.pl",
    9 "PGbasicmacros.pl",
   10 "PGchoicemacros.pl",
   11 "PGanswermacros.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