{VERSION 4 0 "IBM INTEL NT" "4.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{PSTYLE "Normal " -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 3" -1 5 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 1 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 12 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple P lot" -1 13 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 1 1 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 257 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {PARA 4 "" 0 "" {TEXT -1 33 "Module 10 : Serious About Series " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 3 "" 0 "" {TEXT -1 27 "1001 \+ : Series & Convergence" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 " " {TEXT -1 17 "O B J E C T I V E" }}{PARA 0 "" 0 "" {TEXT -1 197 "In t his module, we will examine sequences and series from both a numerical and geometric point of view. We will look at convergence of sequence \+ and series, and several convergence tests for series." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 9 "S E T U P" }}{PARA 0 "" 0 "" {TEXT -1 253 "In this project we will use the following comman d packages. Type and execute this line before beginning the project be low. If you re-enter the worksheet for this project, be sure to re-exe cute this statement before jumping to any point in the worksheet." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 5 "" 0 "" {TEXT -1 83 "_____________________________________________________________________ ______________" }}{PARA 4 "" 0 "" {TEXT -1 32 "A. Sequences Numeric & \+ Geometric" }}{PARA 0 "" 0 "" {TEXT -1 83 "____________________________ _______________________________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 311 "In a sense, a sequence is the range of a function define d only on the positive whole numbers. A function can also be thought o f as a numbers. Here we define the function a which is only defined fo r non-negative values of k. There are two ways to write ouit the seque nce using the $ operator and the seq command." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 33 "defining a sequence as a \+ function" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "a := k -> ((-1)^k) * (2^k - 1)/(k! + 2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aGR6#%\"kG6\"6$%)operatorG%&arrowGF(*&*&)! \"\"9$\"\"\",&)\"\"#F0F1F1F/F1F1,&-%*factorialG6#F0F1F4F1F/F(F(F(" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 "listing t he first 12 elements" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "a(k) $ k = 1..12;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6.#!\"\"\"\"$#F%\"\"%#!\"(\"\")#\"#:\"#E#!#J\"$A\"#\"#j\" $A(#!$F\"\"%U]#\"$b#\"&A.%#!$6&\"'#)GO#\"%B5\"(-)GO#!%Z?\")-o\"*R#\"%& 4%\"*-;+z%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 "...in two different ways" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "seq( a(k), k = 1..12);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6.#!\"\"\"\"$#F%\"\"%#!\"(\"\")#\"#:\"#E #!#J\"$A\"#\"#j\"$A(#!$F\"\"%U]#\"$b#\"&A.%#!$6&\"'#)GO#\"%B5\"(-)GO#! %Z?\")-o\"*R#\"%&4%\"*-;+z%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 40 "expressing the numbers in decimal format" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "seq( evalf(a(k)), k = 1..12);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6.$ !+LLLLL!#5$\"+++++vF%$!++++]()F%$\"+p2BpdF%$!+2O)4a#F%$\"+tG!#8$!+Ij;G^!#9$\"+^>.\\&)!#:" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 106 "We can also see what the limit of the sequence is, if the limit exists, and \+ plot the sequence graphically." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "Limit( a(k), k = infinity): \+ % = value(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&LimitG6$*&*&)!\" \"%\"kG\"\"\",&)\"\"#F+F,F,F*F,F,,&-%*factorialG6#F+F,F/F,F*/F+%)infin ityG\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "plot( a(floo r(x)), x = 1..30 );" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6%-%'CURVESG6$7br7$$\"\"\"\"\"!$!3:LLLLLLLL!#=7$$\"3jmmTN (egJ\"!#F 1F+7$$\"3eX'GNh$\\T>F1F+7$$\"3W\"z%\\S(oe(>F1F+7$$\"3Qky(RIcI*>F1F+7$$ \"3JP4YnQC5?F1$\"3++++++++vF-7$$\"3C5S%4VJu-#F1FG7$$\"3<$3FW**=Y/#F1FG 7$$\"3!\\Pf$[#pL6#F1FG7$$\"3im;H-&>@=#F1FG7$$\"3c*\\Pf!pQ\"\\#F1FG7$$ \"3&HL$e4Vl+GF1FG7$$\"3!H$3xmO[yGF1FG7$$\"3%GLeR-8j&HF1FG7$$\"3%z?bKOq d(HF1FG7$$\"3/$3_DqF_*HF1FG7$$\"39e*[=/&o9IF1$!3+++++++]()F-7$$\"3BLe9 \"QUT.$F1Fdo7$$\"3U$eR(fq0tIF1Fdo7$$\"3=LLLQ<(>6$F1Fdo7$$\"3_L$3FXIwE$ F1Fdo7$$\"3ULL3n\"*GBMF1Fdo7$$\"3[;/wi!3#yNF1Fdo7$$\"3)**\\P%ep7LPF1Fd o7$$\"3&>/wiS'e5QF1Fdo7$$\"3[$e9T&e/))QF1Fdo7$$\"3CaQ.ybxERF1Fdo7$$\"3 -DJ&>I0b'RF1Fdo7$$\"3=gF\"R;q[)RF1Fdo7$$\"3L&Rse-NU+%F1$\"3so2Bp2BpdF- 7$$\"3$4.Ky))*fBSF1F]r7$$\"3`m;z\\Z'H/%F1F]r7$$\"3(***\\PWSAIVF1F]r7$$ \"3TL$e*QL[N?mZF1F]r7$$\"3Km;z*pB\\\"\\F1F]r7$$\" 3;Le*[u.@&\\F1F]r7$$\"3++++!z$G*)\\F1F]r7$$\"3T$3_D\"Q(y+&F1$!34xtb1O) 4a#F-7$$\"3$o;/^$QYE]F1Fjs7$$\"3O\\ildQ0X]F1Fjs7$$\"3yK$3-)Qkj]F1Fjs7$ $\"3dlmTqR+Q^F1Fjs7$$\"3C**\\igSO7_F1Fjs7$$\"3pm;H2t(*>bF1Fjs7$$\"3DL$ eRb!fFeF1Fjs7$$\"3xl\"HdBZU!fF1Fjs7$$\"3=****\\7$$\"3(eT&QeAB>gF1Fju7$$\"3xB\"G)GkRQgF1Fju7$$ \"3cK3F*fgv0'F1Fju7$$\"3E\\i:S*))e4'F1Fju7$$\"3(fmT5G0(F1Fgx7$$\"3Emm;k0lrqF1Fgx7$$\"3c**\\7V zX\\tF1Fgx7$$\"3'GL$3A`EFwF1Fgx7$$\"3]*\\(=xzj$y(F1Fgx7$$\"3.n;HK1,SzF 1Fgx7$$\"3a(ozT@d&fzF1Fgx7$$\"3#*3x1'z.\"zzF1Fgx7$$\"3IId&zP]')*zF1Fgx 7$$\"3!3vV)fp>=!)F1$\"3Cp8\"p1\"4Cj!#?7$$\"3e$z>O7!Hd!)F1Fjz7$$\"3cMeR (G$Q'4)F1Fjz7$$\"3c;z%\\hpX<)F1Fjz7$$\"3M++]Ufv_#)F1Fjz7$$\"3-+++:aym& )F1Fjz7$$\"3q****\\()[\"3)))F1Fjz7$$\"3W++]dg1'[*F1$!3sQ2E^7<39F\\[l7$ $\"3FLeR5#pN+\"!#;$\"3Hi-(G?7\">G!#@7$$\"3ammTj]#*o5Fg\\lFh\\l7$$\"3hm mmV+HC6Fg\\l$!3vfrbHj;G^!#A7$$\"3)**\\ivn#p)=\"Fg\\lFa]l7$$\"3Wmm;@**p X7Fg\\l$\"3w;o$3&>.\\&)!#B7$$\"3\"**\\iD!fC38Fg\\l$!3&3e^-G'R:8F\\^l7$ $\"3/+vonW!yO\"Fg\\lF`^l7$$\"3um;H#eZ*H9Fg\\l$\"3G^HB^ADz=!#C7$$\"3im \"z/O9q[\"Fg\\lFh^l7$$\"3EL$3-fo&[:Fg\\l$!3?q:+Shu0D!#D7$$\"3*H$eRNh]7 ;Fg\\l$\"3#fwz(p/BKJ!#E7$$\"3s*\\Pk&R;o;Fg\\lFg_l7$$\"3ZL$ez#fFGVy80]o$!#F7$$\"3y****\\wuP!z\"Fg\\lF``l7$$\"3%***\\7E<8^=Fg\\l$ \"3)H5mBxlW4%!#G7$$\"3v*\\ile9*4>Fg\\l$!3NGpa\">s*4V!#H7$$\"3q**\\(3$H =v>Fg\\lF_al7$$\"3Hmm;T%HQ.#Fg\\l$\"3J`X'=Iw*4V!#I7$$\"3!)***\\(3rW'4# Fg\\lFhal7$$\"3?LeRA!)=`@Fg\\l$!3=1KS#HTZ5%!#J7$$\"3?++DG3A:AFg\\l$\"3 I\\Hx#)QeJP!#K7$$\"3Em\"HU'))etAFg\\lFgbl7$$\"3%**\\7)z-gMBFg\\l$!3pTV 0Pf&[C$!#L7$$\"3UmmT!4]UR#Fg\\lF`cl7$$\"3y*\\iS\"zpcCFg\\l$\"3vPdQvn// F!#M7$$\"3ELLL!fUo^#Fg\\l$!3/'4yZ[PK;#!#N7$$\"3;L$3(3#\\$yDFg\\lF_dl7$ $\"3gm\"H7_Y$REFg\\l$\"3a\\1Cf)GSm\"!#O7$$\"3#)****\\;nR&p#Fg\\lFhdl7$ $\"3am;alljfFFg\\l$!3J(e\"**yRhK7!#P7$$\"3ILLL_M42'!# S7$$\"#IF*$\"3=1AD*Q$*z/%!#T-%'COLOURG6&%$RGBG$\"#5!\"\"$F*F*F`gl-%+AX ESLABELSG6$Q\"x6\"Q!6\"-%%VIEWG6$;F(Fdfl%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 5 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 5 "" 0 "" {TEXT -1 83 "_____________________________________________________________________ ______________" }}{PARA 4 "" 0 "" {TEXT -1 23 "B. Series & Convergence " }}{PARA 0 "" 0 "" {TEXT -1 83 "_____________________________________ ______________________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 50 "If we add the terms in a sequence we get a series." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 17 "FINITE & INFINIT E" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 100 "We \+ can add up a finite or infinite number of terms in a sequence to get a finite or infinite series." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 138 "A sum can contain a finite or infinite number \+ of terms. Even when the sum is adding an infinite number of terms, the result can be finite." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "Sum( 3*k^4 - 5*k, k = 1..30): % = value(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$SumG6$,&*$)%\"kG\"\" %\"\"\"\"\"$*&\"\"&F,F*F,!\"\"/F*;F,\"#I\")s'>e\"" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 48 "Sum( k/10000, k = 1..infinity): % = value (%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$SumG6$,$%\"kG#\"\"\"\"&++ \"/F(;F*%)infinityGF." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "Su m( 1/k^6, k = 1..infinity): % = value(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$SumG6$*&\"\"\"F(*$)%\"kG\"\"'F(!\"\"/F+;F(%)infinit yG,$*$)%#PiGF,F(#F(\"$X*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "Sum( 1/k, k = 1..infinity): % = value(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$SumG6$*&\"\"\"F(%\"kG!\"\"/F);F(%)infinityGF-" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 24 "SEQUENC E OF PARTIAL SUMS" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 293 "Every infinite series can be thought of as a an infinite sequence. Let S1 = a1, S2 = a1 + a2, S3 = a1 + a2 +a3 ,...etc. Each S n is called a \"partial sum\" because it made up of only the sum of th e first n terms. The limit of the sequence of partial sums \{Sn\} is t he same as the infinite series." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "a := k -> 1/k^(3/2);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aGR6#%\"kG6\"6$%)operatorG%&arrowG F(*&\"\"\"F-*$)9$#\"\"$\"\"#F-!\"\"F(F(F(" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 31 "S := n -> sum( a(k), k = 1..n);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%\"SGR6#%\"nG6\"6$%)operatorG%&arrowGF(-%$sumG6$-%\" aG6#%\"kG/F2;\"\"\"9$F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "array([seq( [ 5*k, evalf( a(5*k) ), evalf( sum( a(j), j = 1..(5*k ))) ], k =1..12)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7.7 %\"\"&$\"+7>FW*)!#6$\"+*>Y/w\"!\"*7%\"#5$\"+gwFiJF+$\"+%\\O`*>F.7%\"#: $\"+JfK@1e3'FE$\"+FXC]AF.7%\"#N$\"+%)GXH[FE$\"+G6rwAF.7 %\"#S$\"+vq%G&RFE$\"+b;6)H#F.7%\"#X$\"++Lp7LFE$\"+(4!)eJ#F.7%\"#]$\"+D rUGGFE$\"+z(R4L#F.7%\"#b$\"+jej^CFE$\"+]c\"RM#F.7%\"#g$\"+:ul^@FE$\"+/ yCbBF." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 194 "The first column is k which increases in increments of 5. The sec ond column is ak, and the third is Sk. Note that the ak's are approach ing 0, while the Sk appear to approach some son-zero limit." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 122 "To help make t he difference between a sequence and its corresponding series more app arent , lets look at both graphically." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "plot( \{ a(floor(x)), su m( a(k), k = 1..floor(x) ) \}, x = 1..30);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6$7[s7$$\"\"\"\"\"!F(7$ $\"3jmmTN(egJ\"!#F.F(7$$\"3eX'GNh$\\T>F.F(7$$\"3W\"z%\\S(oe(>F.F(7$$\"3Qky(RIc I*>F.F(7$$\"3JP4YnQC5?F.$\"3'ytKf!R`NN!#=7$$\"3C5S%4VJu-#F.FD7$$\"3<$3 FW**=Y/#F.FD7$$\"3!\\Pf$[#pL6#F.FD7$$\"3im;H-&>@=#F.FD7$$\"3c*\\Pf!pQ \"\\#F.FD7$$\"3&HL$e4Vl+GF.FD7$$\"3!H$3xmO[yGF.FD7$$\"3%GLeR-8j&HF.FD7 $$\"3%z?bKOqd(HF.FD7$$\"3/$3_DqF_*HF.FD7$$\"39e*[=/&o9IF.$\"3M_()H(*3] C>FF7$$\"3BLe9\"QUT.$F.Fbo7$$\"3U$eR(fq0tIF.Fbo7$$\"3=LLLQ<(>6$F.Fbo7$ $\"3_L$3FXIwE$F.Fbo7$$\"3ULL3n\"*GBMF.Fbo7$$\"3[;/wi!3#yNF.Fbo7$$\"3)* *\\P%ep7LPF.Fbo7$$\"3&>/wiS'e5QF.Fbo7$$\"3[$e9T&e/))QF.Fbo7$$\"3CaQ.yb xERF.Fbo7$$\"3-DJ&>I0b'RF.Fbo7$$\"3=gF\"R;q[)RF.Fbo7$$\"3L&Rse-NU+%F.$ \"3+++++++]7FF7$$\"3$4.Ky))*fBSF.F[r7$$\"3`m;z\\Z'H/%F.F[r7$$\"3(***\\ PWSAIVF.F[r7$$\"3TL$e*QL[N?mZF.F[r7$$\"3Km;z*pB\\ \"\\F.F[r7$$\"3;Le*[u.@&\\F.F[r7$$\"3++++!z$G*)\\F.F[r7$$\"3T$3_D\"Q(y +&F.$\"3Ag\"***4>FW*)!#>7$$\"3$o;/^$QYE]F.Fhs7$$\"3O\\ildQ0X]F.Fhs7$$ \"3yK$3-)Qkj]F.Fhs7$$\"3dlmTqR+Q^F.Fhs7$$\"3C**\\igSO7_F.Fhs7$$\"3pm;H 2t(*>bF.Fhs7$$\"3DL$eRb!fFeF.Fhs7$$\"3xl\"HdBZU!fF.Fhs7$$\"3=****\\gF.Fiu7$$ \"3xB\"G)GkRQgF.Fiu7$$\"3cK3F*fgv0'F.Fiu7$$\"3E\\i:S*))e4'F.Fiu7$$\"3( fmT5G0(F.Fex7$$\"3Em m;k0lrqF.Fex7$$\"3c**\\7VzX\\tF.Fex7$$\"3'GL$3A`EFwF.Fex7$$\"3]*\\(=xz j$y(F.Fex7$$\"3.n;HK1,SzF.Fex7$$\"3a(ozT@d&fzF.Fex7$$\"3#*3x1'z.\"zzF. Fex7$$\"3IId&zP]')*zF.Fex7$$\"3!3vV)fp>=!)F.$\"3LAfT#Q<%>WFjs7$$\"3e$z >O7!Hd!)F.Fhz7$$\"3cMeR(G$Q'4)F.Fhz7$$\"3c;z%\\hpX<)F.Fhz7$$\"3M++]Ufv _#)F.Fhz7$$\"3-+++:aym&)F.Fhz7$$\"3q****\\()[\"3)))F.Fhz7$$\"3&)**\\i& 3V'=*)F.Fhz7$$\"3C)**\\PGrk&*)F.Fhz7$$\"3W(\\7GQ&Qv*)F.Fhz7$$\"3S)*\\( =[*H%**)F.Fhz7$$\"3O*\\P4e8K,*F.$\"3].Pq.Pq.PFjs7$$\"3b)*****zw7K!*F.F []l7$$\"3'))**\\i2%y2\"*F.F[]l7$$\"3;****\\s/W$=*F.F[]l7$$\"3!)*****\\ E`ZL*F.F[]l7$$\"3W++]dg1'[*F.F[]l7$$\"3FLeR5#pN+\"!#;$\"3%)z$o,mxA;$Fj s7$$\"3ammTj]#*o5F_^lF`^l7$$\"3hmmmV+HC6F_^l$\"3%[@MMA75u#Fjs7$$\"3)** \\ivn#p)=\"F_^lFh^l7$$\"3Wmm;@**pX7F_^l$\"3VSMi@hi0CFjs7$$\"3\"**\\iD! fC38F_^l$\"3CeR<$HiM8#Fjs7$$\"3/+vonW!yO\"F_^lFe_l7$$\"3um;H#eZ*H9F_^l $\"3JJI!3()3!4>Fjs7$$\"3im\"z/O9q[\"F_^lF]`l7$$\"3EL$3-fo&[:F_^l$\"3%3 uZ;$fK@-qX48Fjs7$$\"3v*\\ile9*4>F_^l$\"3M$p(*3B^u? \"Fjs7$$\"3q**\\(3$H=v>F_^lF_bl7$$\"3Hmm;T%HQ.#F_^l$\"3.&*)\\())R.=6Fj s7$$\"3!)***\\(3rW'4#F_^lFgbl7$$\"3?LeRA!)=`@F_^l$\"3l#eZ1\"G8R5Fjs7$$ \"3?++DG3A:AF_^l$\"3#puFDlT4p*!#?7$$\"3Em\"HU'))etAF_^lFdcl7$$\"3%**\\ 7)z-gMBF_^l$\"3OL!Q%*3We1*Ffcl7$$\"3UmmT!4]UR#F_^lF]dl7$$\"3y*\\iS\"zp cCF_^l$\"3>Yr*zrs^])Ffcl7$$\"3ELLL!fUo^#F_^l$\"3=+++++++!)Ffcl7$$\"3;L $3(3#\\$yDF_^lFjdl7$$\"3gm\"H7_Y$REF_^l$\"3\"*Rbau#GHa(Ffcl7$$\"3#)*** *\\;nR&p#F_^lFbel7$$\"3am;alljfFF_^l$\"32!\\165\"yFrFfcl7$$\"3ILLL_M4< GF_^l$\"3=&[]%*el$\\nFfcl7$$\"3)**\\ivZa$yGF_^lF_fl7$$\"3%)*\\Pa4*)p$H F_^l$\"3u:mMBvG.kFfcl7$$\"#IF*$\"3AX=]%>1e3'Ffcl-%'COLOURG6&%$RGBG$\"# 5!\"\"$F*F*Fegl-F$6$7c[lF'F+F/F2F5F8F;F>7$FB$\"3stKf!R`NN\"F.7$FHFjgl7 $FKFjgl7$FNFjgl7$FQFjgl7$FTFjgl7$FWFjgl7$FZFjgl7$FgnFjgl7$FjnFjgl7$F]o Fjgl7$F`o$\"31\\JK![.ga\"F.7$FeoFghl7$FhoFghl7$F[pFghl7$F^pFghl7$FapFg hl7$FdpFghl7$FgpFghl7$FjpFghl7$F]qFghl7$F`qFghl7$FcqFghl7$FfqFghl7$Fiq $\"3/\\JK![.5n\"F.7$F^rFfil7$FarFfil7$FdrFfil7$FgrFfil7$FjrFfil7$F]sFf il7$F`sFfil7$FcsFfil7$Ffs$\"3iSJU*>Y/w\"F.7$F\\tFajl7$F_tFajl7$FbtFajl 7$FetFajl7$FhtFajl7$F[uFajl7$F^uFajl7$FauFajl7$FduFajl7$Fgu$\"3uF.7$F[[lFj\\m7$ F^[lFj\\m7$Fa[lFj\\m7$Fd[lFj\\m7$Fg[lFj\\m7$Fj[lFj\\m7$F]\\lFj\\m7$F` \\lFj\\m7$Fc\\lFj\\m7$Ff\\lFj\\m7$Fi\\l$\"3#y\"Ru;Prj>F.7$F^]lFg]m7$Fa ]lFg]m7$Fd]lFg]m7$Fg]lFg]m7$Fj]lFg]m7$$\"3g#e9\"pDZB'*F.Fg]m7$$\"3am\" H23z3w*F.Fg]m7$$\"3jdk`OBeH)*F.Fg]m7$$\"3[]PM#f&G)*)*F.Fg]m7$$\"3#pRZ- APE$**F.Fg]m7$$\"3eT5:[)))p'**F.Fg]m7$$\"3-jG5iY;%)**F.Fg]m7$$\"3ioagZ S8+5F_^l$\"3q,cM$\\O`*>F.7$$\"3&4l+!H;&=+\"F_^lFf_m7$F]^lFf_m7$$\"3!* \\i!p8Zi.\"F_^lFf_m7$Fc^lFf_m7$$\"3km\"z%3jw#3\"F_^lFf_m7$$\"3em;a`vg' 4\"F_^lFf_m7$$\"3AzW<4xL)4\"F_^lFf_m7$$\"3o\"H2['y1+6F_^l$\"31W*zbhYF- #F.7$$\"3:/,W?!)z,6F_^lF\\am7$$\"3i;H2w\"GN5\"F_^lF\\am7$$\"3cT&Qt[))p 5\"F_^lF\\am7$$\"3]mTg)z[/6\"F_^lF\\am7$$\"3b;a8@%pt6\"F_^lF\\am7$Ff^l F\\am7$$\"3I$e91O\"\\c6F_^lF\\am7$F[_lF\\am7$$\"3SgxGIcD#>\"F_^lF\\am7 $$\"3%3-8Ie=e>\"F_^lF\\am7$$\"3)4lv$f+g(>\"F_^lF\\am7$$\"35\"GQd`\"Q*> \"F_^lF\\am7$$\"3B6457I;,7F_^l$\"3MyhzwG!o/#F.7$$\"3aTNY)[WH?\"F_^lFac m7$$\"3CiS\"RRq+@\"F_^lFacm7$$\"36$ek$*H'><7F_^lFacm7$$\"3([il-6[9B\"F _^lFacm7$F^_lFacm7$$\"3'[i:lTO8E\"F_^lFacm7$$\"33$ek=\"H(pF\"F_^lFacm7 $$\"3Qi!R&f6z%G\"F_^lFacm7$$\"3]TN@2%4EH\"F_^lFacm7$$\"3Kh@8pRc%H\"F_^ lFacm7$$\"3'4y]5`=lH\"F_^lFacm7$$\"3!3SpH4t%)H\"F_^lFacm7$$\"3h?!))[lF /I\"F_^l$\"3#y\"zs*\\P\"o?F.7$$\"3Eg_synL/8F_^lFhem7$Fc_lFhem7$$\"3)** *\\7&=D!Q8F_^lFhem7$Fh_lFhem7$$\"3oT&QjCSLQ\"F_^lFhem7$$\"3I$e*)\\-w)) R\"F_^lFhem7$$\"3dj4K(*z\"3S\"F_^l$\"3O[fV)eFs3#F.7$$\"3%QM_'p*fFS\"F_ ^lF[gm7$$\"36CP)>%>q/9F_^lF[gm7$$\"3@/^J9Rk19F_^lF[gm7$$\"3dky(*ey_59F _^lF[gm7$$\"36D1k.=T99F_^lF[gm7$$\"3%e9mHpz@U\"F_^lF[gm7$F[`lF[gm7$$\" 3o;aQr4[e9F_^lF[gm7$F``lF[gm7$$\"3kRs$)*\\h3\\\"F_^lF[gm7$$\"3[7`>R'3Z \\\"F_^lF[gm7$$\"3!*[V()3Aj'\\\"F_^lF[gm7$$\"3K&Q`&ydb)\\\"F_^lF[gm7$$ \"3t@CB[$z/]\"F_^l$\"3!fU_x%3W/@F.7$$\"3:e9\"z\"HS-:F_^lFcim7$$\"3+/wi '>(45:F_^lFcim7$$\"3&)\\PMv9z<:F_^lFcim7$$\"3cTgxK+=L:F_^lFcim7$Fc`lFc im7$$\"3?3_]wHbk:F_^lFcim7$$\"39$3-GOP0e\"F_^lFcim7$$\"3g?0&fbH&)e\"F_ ^lFcim7$$\"31e*)4\\<_'f\"F_^lFcim7$$\"3_ngQ(z>&)f\"F_^lFcim7$$\"3'p>$o\"F_^lFd[n7$$\"3w;z>U*>#)p\"F_^lFd[n7$$\"3#H%zIT%)4 +q\"F_^lF]]n7$$\"3B&*z_Ra&Qq\"F_^lF ]]n7$$\"3.@!Q'QRt0w?sUZ@F.7$$\"37oagoum.=F_^lF b_n7$$\"3c\\i!*Qgc0=F_^lFb_n7$$\"3ju$4,KgJ\"=F_^lFb_n7$$\"3p*\\78ga2#= F_^lFb_n7$$\"3#)\\(=P;Vf$=F_^lFb_n7$FhalFb_n7$$\"33vVBTu#e'=F_^lFb_n7$ $\"3')\\PMcJ_!)=F_^lFb_n7$$\"3UP%)*Q,ry)=F_^lFb_n7$$\"3iCJXr)=_*=F_^lF b_n7$$\"3U'zTe$e0(*=F_^lFb_n7$$\"3Ao/B+G*))*=F_^lFb_n7$$\"3QS\">YwH2!> F_^l$\"39,hbJ<]f@F.7$$\"3>7y+Hnc->F_^lFian7$$\"3:c^yd1C1>F_^lFian7$F]b lFian7$$\"3t\\(=(e([D%>F_^lFian7$FbblFian7$$\"3)zXhYu8D)>F_^lFian7$$\" 3E;zWeX%)*)>F_^lFian7$$\"3eX6Ml*4N*>F_^lFian7$$\"3auVBs`<(*>F_^lFian7$ $\"3%)))4ov!3!**>F_^lFian7$$\"3].w7z2%3+#F_^l$\"39+OWr?oq@F.7$$\"3;=Ud #[tE+#F_^lFhcn7$$\"3\"G$3-'=1X+#F_^lFhcn7$$\"3P\\Pf8y;>?F_^lFhcn7$Febl Fhcn7$$\"3/L$e\\FQ^1#F_^lFhcn7$FjblFhcn7$$\"3QK9Ii-A)4#F_^lFhcn7$$\"3e kG&eT$***4#F_^lFhcn7$$\"3!oH/%plw,@F_^l$\"3;w+b*Rt5=#F.7$$\"3-Hd&HsRN5 #F_^lFaen7$$\"3X$fe+.'32@F_^lFaen7$$\"3)yXhrLK16#F_^lFaen7$$\"35(=n8& \\s<@F_^lFaen7$$\"3L;Hdlv\"[7#F_^lFaen7$$\"3xuV)Rz-!R@F_^lFaen7$F]clFa en7$$\"3/v$fQA'po@F_^lFaen7$$\"3)o\"HKDW?%=#F_^lFaen7$$\"3!yoag_e>>#F_ ^lFaen7$$\"3sekyEEr*>#F_^lFaen7$$\"3N,%p>:^;?#F_^l$\"35/E?TVw!>#F.7$$ \"3+WB:x'*e.AF_^lFegn7$$\"3k'GNB?Gb?#F_^lFegn7$$\"3GH#=vsmu?#F_^lFegn7 $$\"3c9T)yxV8@#F_^lFegn7$FbclFegn7$$\"3T$eRi%[SWAF_^lFegn7$FhclFegn7$$ \"31$3x6H:7G#F_^lFegn7$$\"3')**\\7=<%))G#F_^lFegn7$$\"3Ee*)fJ\\l#H#F_^ lFegn7$$\"3m;H2X\"okH#F_^lFegn7$$\"3'e*)4=vu$)H#F_^lFegn7$$\"30voae8G+ BF_^l$\"35U?H&=I)*>#F.7$$\"3DaQGlz=-BF_^lFjin7$$\"35L3-sX4/BF_^lFjin7$ $\"3Mmm\"fUZ$>BF_^lFjin7$F[dlFjin7$$\"3=$e9^=DWO#F_^lFjin7$F`dlFjin7$F cdl$\"3LR+,e`L3AF.7$Fhdl$\"3MR+,e`L;AF.7$F]elF^[o7$F`el$\"3&\\e%G'GyQA #F.7$FeelFb[o7$Fhel$\"3_\"p&Qng+JAF.7$F]fl$\"3!>9vHVbxB#F.7$FbflFi[o7$ Fefl$\"3`)[)\\?(eTC#F.7$Fjfl$\"3Y!*HpEXC]AF.-F_gl6&FaglFeglFbglFegl-%+ AXESLABELSG6$Q\"x6\"Q!6\"-%%VIEWG6$;F(Fjfl%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 96 "The terms of the sequence are decreasing to zero while the partial sums converg e to some number." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 5 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 5 "" 0 "" {TEXT -1 83 "_______________________________ ____________________________________________________" }}{PARA 4 "" 0 " " {TEXT -1 18 "C. Slow Divergence" }}{PARA 0 "" 0 "" {TEXT -1 83 "____ ______________________________________________________________________ _________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 253 "Each when a series diverges to in finity, the journey may be quite leisurely. Lets examine the harmonic \+ series, which is the sum of the reciprocals of the natural numbers. Fi rst we'll add up the first 100 reciprocals and see how close we are to infinity." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 34 "Sum( 1/k, k = 1..100); value(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*&\"\"\"F'%\"kG!\"\"/F(;F'\"$+\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6##\"J6xWJT5V!=:Ag6N?&zijmW\"\"IsA9#\\7u N_8e'3*\\)=4]\"))y#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 4 "The " }{TEXT 256 6 "evalf " }{TEXT -1 32 "command forces a decimal answer." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(% );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+=vP(=&!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 78 "Not too close! Let add up the first 1,000 terms and see what progress is made." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "Sum( \+ 1/k, k = 1..1000); % = evalf( value(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*&\"\"\"F'%\"kG!\"\"/F(;F'\"%+5" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/-%$SumG6$*&\"\"\"F(%\"kG!\"\"/F);F(\"%+5$\"+h3Z &[(!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 105 "disappointingly small! Surely by the time we add a million terms, we must be getting significantly large!" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "Sum( 1/k, k = 1..1000 000); % = evalf( value(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$Su mG6$*&\"\"\"F'%\"kG!\"\"/F(;F'\"(+++\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$SumG6$*&\"\"\"F(%\"kG!\"\"/F);F(\"(+++\"$\"+sEFR9!\")" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "0 0" 0 } {VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }