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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT 256 45 "/home/m262f99/KOEPF/works
heetsV.4/hw1ansm.mws" }{MPLTEXT 1 0 0 "" }}{PARA 258 "" 0 "" {TEXT -1 
45 "Math 262a, Fall 1999, Glenn Tesler\nHomework 1" }{MPLTEXT 1 0 0 "
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "read `hsum.mpl`;" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#%ZCopyright~1998~~Wolfram~Koepf,~Konrad-Zuse-Zentrum~
BerlinG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "To get help on the pac
kage, type" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "?hsum" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 36 "Problem 1. Koepf 2.11(b)\nTerm ratio:" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 27 "ratio((-1)^(k+1)/k*x^k, k);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,$*(,&%\"kG\"\"\"F'F'!\"\"%\"xGF'F&F'F(" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 10 "whole sum:" }}{PARA 0 "> " 0 "" {MPLTEXT 
1 0 31 "Sumtohyper((-1)^(k+1)/k*x^k,k);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#*&%\"xG\"\"\"-%*HypergeomG6%7$F%F%7#\"\"#,$F$!\"\"F%" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 28 "maple's built-in sum command" }}{PARA 0 "
> " 0 "" {MPLTEXT 1 0 41 "sum((-1)^(k+1)/k*x^k,k); # indefinite sum" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$sumG6$*()!\"\",&%\"kG\"\"\"F+F+F+F
*F()%\"xGF*F+F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "sum((-1)
^(k+1)/k*x^k,k=1..infinity); # definite sum" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%#lnG6#,&\"\"\"F'%\"xGF'" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 36 "sum((-1)^(k+1)/k*x^k,k=5..infinity);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#,$*&%\"xG\"\"&,,*$F%!\"\"#F&\"\"%*$F%!\"##!\"&\"\"
$*$F%!\"$#F&\"\"#*$F%!\"%F/*&F%F/-%#lnG6#,&\"\"\"F<F%F<F<F&F<#F<F&" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(\");" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#,,*$%\"xG\"\"%#\"\"\"F&*$F%\"\"$#!\"\"F**$F%\"\"
##F(F.F%F,-%#lnG6#,&F(F(F%F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 13 "Koepf 2.11(f)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 117 "atanterm :
= (-1)^k/(2*k+1) * x^(2*k+1);\nsubs(k=0,atanterm); # initial term\nrat
io(atanterm,k);\nSumtohyper(atanterm,k);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%)atantermG*()!\"\"%\"kG\"\"\",&F(\"\"#F)F)F')%\"xGF*F)" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#%\"xG" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#,$*(,&%\"kG\"\"#\"\"$\"\"\"!\"\"%\"xGF',&F&F'F)F)F)F*" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#*&%\"xG\"\"\"-%*HypergeomG6%7$#F%\"\"#F%7##\"\"$
F+,$*$F$F+!\"\"F%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "maple's buil
t-in sum" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "sum(atanterm,k); # inde
finite sum\nsum(atanterm,k=0..infinity); # definite sum" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#-%$sumG6$*()!\"\"%\"kG\"\"\",&F)\"\"#F*F*F()%\"x
GF+F*F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(%\"xG\"\"\",$*$F%\"\"#!
\"\"#F*F)-%#lnG6#*&,&F&F&*$F'#F&F)F&F&,&F&F&F1F*F*F&F2" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 12 "Koepf 2.9(d)
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "dterm := binomial(n,k)*binomial
(2*k,n);\nratio(dterm,k);\nSumtohyper(dterm,k);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%&dtermG*&-%)binomialG6$%\"nG%\"kG\"\"\"-F'6$,$F*\"\"#
F)F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$**,&%\"nG!\"\"%\"kG\"\"\"F),
&F(\"\"#F)F)F),(F(F+F&F'F)F)F',(F(F+F+F)F&F'F'!\"#" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#*&-%)binomialG6$\"\"!%\"nG\"\"\"-%*HypergeomG6%7%#F)
\"\"#,$F(!\"\"F)7$,&F)F)F(#F1F/,&F(F4F.F)F1F)" }}}{EXCHG {PARA 0 "" 0 
"" {TEXT -1 104 "Note: the above answer assumes n is not a nonnegative
 integer. See the answer key for a complete answer." }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 30 "Sumtohyper(subs(n=5,dterm),k);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#,$-%*HypergeomG6%7$!\"##\"\"(\"\"#7##\"\"$F+
!\"\"\"#g" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "sum(dterm,n=0..infinity);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#*&-%)binomialG6$\"\"!%\"kG\"\"\"-%*hypergeom
G6%7$F),$F(!\"#7#,&F(!\"\"F)F)F2F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 17 "Here's Method II." }{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 67 "dterm2 := subs(k=n-k,dterm);\nratio(dterm2,k);\nSumto
hyper(dterm2,k);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'dterm2G*&-%)bin
omialG6$%\"nG,&F)\"\"\"%\"kG!\"\"F+-F'6$,&F)\"\"#F,!\"#F)F+" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#,$**,&%\"kG\"\"#%\"nG!\"\"\"\"\",(F&F'F*F*F(
F)F*,(F(!\"#F&F'F*F*F),&F&F*F*F*F)#F)F'" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#*&-%)binomialG6$,$%\"nG\"\"#F(\"\"\"-%*HypergeomG6%7$,&#F*F)F*F(
#!\"\"F),$F(F17#,&F(F2F0F*F2F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "Problems 2-3 aren't he
re." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 21 "Problem 4, Koepf 2.21" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "read `qsum.mpl`;" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#%OCopyright~1998,~~Harald~Boeing~&~Wolfram~KoepfG" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#%;Konrad-Zuse-Zentrum~BerlinG" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 7 "2.21(a)" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 67 "qsimplify(qpochhammer(a,q,infinity)/qpochhammer(a*q^n
,q,infinity));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%,qpochhammerG6%%\"
aG%\"qG%\"nG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 40 "2.21(b)  the symb
ol b stands for sqrt(a)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "qsimplif
y(qpochhammer(q*b,q,n)*qpochhammer(-q*b,q,n) / (qpochhammer(b,q,n)*qpo
chhammer(-b,q,n)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#**,&!\"\"\"\"\"
*&%\"bGF&)%\"qG%\"nGF&F&F&,&F&F&F'F&F&,&F%F&F(F&F%,&F&F&F(F&F%" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "expand(numer(\"))/expand(den
om(\"));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&!\"\"\"\"\"*&%\"bG\"\"
#)%\"qG%\"nGF)F&F&,&F%F&*$F(F)F&F%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 7 "2.21(c)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "qsimplify(qpochham
mer(a,q,n)*qpochhammer(-a,q,n));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%
,qpochhammerG6%*$%\"aG\"\"#*$%\"qGF(%\"nG" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 7 "2.21(d)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 93 "form1 := qpo
chhammer(a,q,n);\nform2 := qpochhammer(q^(1-n)/a,q,n);\nform2b := qsim
plify(form2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&form1G-%,qpochhamm
erG6%%\"aG%\"qG%\"nG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&form2G-%,qp
ochhammerG6%*&)%\"qG,&\"\"\"F,%\"nG!\"\"F,%\"aGF.F*F-" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%'form2bG*,)%\"qG,$%\"nG#\"\"\"\"\"#F+)!\"\"F)F+-
%,qpochhammerG6%%\"aGF'F)F+)F',$*$F)F,F*F.)F2F)F." }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 29 "solve(form2=form2b, \{form1\});" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#<#/-%,qpochhammerG6%%\"aG%\"qG%\"nG*,)F),$*$F*
\"\"##\"\"\"F/F1)F(F*F1)F),$F*F0!\"\")F5F*F5-F&6%*&)F),&F1F1F*F5F1F(F5
F)F*F1" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 138 "built-in maple command
 that doesn't know about q functions/simplifications, so it won't touc
h the qpochhammer's but will simplify the rest" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 12 "simplify(\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<#/-
%,qpochhammerG6%%\"aG%\"qG%\"nG**)!\"\",$F*F-\"\"\")F),&*$F*\"\"##F/F3
F*#F-F3F/)F(F*F/-F&6%*&)F),&F/F/F*F-F/F(F-F)F*F/" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "Koepf
 3.15(b)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "sum2qhyper(qbinomial(n,
k,q)^2 * x^k, q, k);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$phiG6&7&)%
\"qG,$%\"nG!\"\"F'\"\"!F,7#F(F(*&)F(,$F*\"\"#\"\"\"%\"xGF2" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "qratio(qbinomial(n,k,q)^2 * x^k, k)
;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#**,&)%\"qG%\"kG!\"\")F&%\"nG\"\"
\"\"\"#%\"xGF+,&F(F+*&F&F+F%F+F+!\"#F%F0" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}}{MARK "0 0 0" 41 }{VIEWOPTS 1 1 0 1 1 1803 }