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0 0 1 1 2 2 2 2 2 0 0 0 1 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT 200 28 "MATH 156 FINAL LAB PROJEC T 1" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 204 52 "In this project you will \+ investigate the sum of the " }{XPPEDIT 18 0 "p;" "6#%\"pG" }{TEXT 204 8 "-series " }{XPPEDIT 18 0 "sum(1/n^p, n = 1 .. infinity);" "6#-%$sum G6$*&\"\"\"F')%\"nG%\"pG!\"\"/F);F'%)infinityG" }{TEXT 204 37 " and u se the results to approximate " }{XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT 204 58 ". We will take p=2, 4, 6, 8, 10, 3, 5 successively. Define" }} {PARA 0 "" 0 "" {TEXT 204 13 "the sequence " }{XPPEDIT 18 0 "1/n^p;" " 6#*&\"\"\"F$)%\"nG%\"pG!\"\"" }{TEXT 204 16 " and the series " } {XPPEDIT 18 0 "sum(1/n^p, n = 1 .. infinity);" "6#-%$sumG6$*&\"\"\"F') %\"nG%\"pG!\"\"/F);F'%)infinityG" }}{PARA 0 "" 0 "" {TEXT 204 0 "" }} {PARA 0 "" 0 "" {TEXT 204 49 "1. Calculate the sequence of partial s ums with " }{XPPEDIT 18 0 "N = 10, 20, 30, etc;" "6&/%\"NG\"#5\"#?\"#I %$etcG" }{TEXT 204 48 ". Use a loop to make the computations.for p=2. \+ " }}{PARA 0 "" 0 "" {TEXT 204 0 "" }}{PARA 0 "" 0 "" {TEXT 204 110 "A n analysis beyond the scope of this course, involving complex variable s shows that the sum of the series for " }{XPPEDIT 18 0 "p = 2;" "6#/% \"pG\"\"#" }{TEXT 204 5 " is " }{XPPEDIT 18 0 "Typesetting:-mrow(Type setting:-mi(\"\"), Typesetting:-mfrac(Typesetting:-mrow(Typesetting:-m i(\"\"), Typesetting:-mverbatim(\")%#PiG\"\"#\"), Typesetting:-mi(\"\" )), Typesetting:-mverbatim(\"*$\"\"'\"\"\"\"), linethickness = \"1\", \+ denomalign = \"center\", numalign = \"center\", bevelled = \"false\", \+ foreground = \"[0,0,0]\", background = \"[255,255,255]\"), Typesetting :-mo(\".\", form = \"infix\", fence = \"false\", separator = \"false\" , lspace = \"0em\", rspace = \"0em\", stretchy = \"false\", symmetric \+ = \"false\", maxsize = \"infinity\", minsize = \"1\", largeop = \"fals e\", movablelimits = \"false\", accent = \"false\", font_style_name = \+ \"2D Comment\", size = 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\"false\", movablelimits = \"false\", accent = \"false\", font_style_name = \"2D Comment\", size = \"12\", foregroun d = \"[0,0,0]\", background = \"[255,255,255]\"), Typesetting:-mi(\"ou tside\"), Typesetting:-mo(\"⁢\", form = \"infix\", fenc e = \"false\", separator = \"true\", lspace = \"0em\", rspace = \"very thickmathspace\", stretchy = \"false\", symmetric = \"false\", maxsize = \"infinity\", minsize = \"1\", largeop = \"false\", movablelimits = \"false\", accent = \"false\", font_style_name = \"2D Comment\", size = \"12\", foreground = \"[0,0,0]\", background = \"[255,255,255]\"), \+ Typesetting:-mi(\"loop\"), Typesetting:-mo(\"⁢\", form \+ = \"infix\", fence = \"false\", separator = \"true\", lspace = \"0em\" , rspace = \"verythickmathspace\", stretchy = \"false\", symmetric = \+ \"false\", maxsize = \"infinity\", minsize = \"1\", largeop = \"false \", movablelimits = \"false\", accent = \"false\", font_style_name = \+ \"2D Comment\", size = \"12\", foreground = \"[0,0,0]\", 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symmetric = \"false\", maxsize = \"infinity\", \+ minsize = \"1\", largeop = \"false\", movablelimits = \"false\", accen t = \"false\", font_style_name = \"2D Comment\", size = \"12\", foregr ound = \"[0,0,0]\", background = \"[255,255,255]\"), Typesetting:-mi( \"the\"), Typesetting:-mo(\"⁢\", form = \"infix\", fenc e = \"false\", separator = \"true\", lspace = \"0em\", rspace = \"very thickmathspace\", stretchy = \"false\", symmetric = \"false\", maxsize = \"infinity\", minsize = \"1\", largeop = \"false\", movablelimits = \"false\", accent = \"false\", font_style_name = \"2D Comment\", size = \"12\", foreground = \"[0,0,0]\", background = \"[255,255,255]\"), \+ Typesetting:-mi(\"inside\"), Typesetting:-mo(\"⁢\", for m = \"infix\", fence = \"false\", separator = \"true\", lspace = \"0em \", rspace = \"verythickmathspace\", stretchy = \"false\", symmetric = \"false\", maxsize = \"infinity\", minsize = \"1\", largeop = \"false \", movablelimits = \"false\", accent = \"false\", font_style_name = \+ \"2D Comment\", size = \"12\", foreground = \"[0,0,0]\", background = \+ \"[255,255,255]\"), Typesetting:-mi(\"loop\"), Typesetting:-mo(\"&Invi sibleTimes;\", form = \"infix\", fence = \"false\", separator = \"true \", lspace = \"0em\", rspace = \"verythickmathspace\", stretchy = \"fa lse\", symmetric = \"false\", maxsize = \"infinity\", minsize = \"1\", largeop = \"false\", movablelimits = \"false\", accent = \"false\", f ont_style_name = \"2D Comment\", size = \"12\", foreground = \"[0,0,0] \", background = \"[255,255,255]\"), Typesetting:-mi(\"computes\"), Ty pesetting:-mo(\"⁢\", form = \"infix\", fence = \"false \", separator = \"true\", lspace = \"0em\", rspace = \"verythickmathsp ace\", stretchy = \"false\", symmetric = \"false\", maxsize = \"infini ty\", minsize = \"1\", largeop = \"false\", movablelimits = \"false\", accent = \"false\", font_style_name = \"2D Comment\", size = \"12\", \+ foreground = \"[0,0,0]\", background = \"[255,255,255]\"), Typesetting 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movablelimits = \"false\", accent = \"false\", font_style_name = \"2D Comment\", size = \"12\", \+ foreground = \"[0,0,0]\", background = \"[255,255,255]\"), Typesetting :-mo(\"⁢\", form = \"infix\", fence = \"false\", separa tor = \"true\", lspace = \"0em\", rspace = \"verythickmathspace\", str etchy = \"false\", symmetric = \"false\", maxsize = \"infinity\", mins ize = \"1\", largeop = \"false\", movablelimits = \"false\", accent = \+ \"false\", font_style_name = \"2D Comment\", size = \"12\", foreground = \"[0,0,0]\", background = \"[255,255,255]\"), Typesetting:-mo(\"&In visibleTimes;\", form = \"infix\", fence = \"false\", separator = \"tr ue\", lspace = \"0em\", rspace = \"verythickmathspace\", stretchy = \" false\", symmetric = \"false\", maxsize = \"infinity\", minsize = \"1 \", largeop = \"false\", movablelimits = \"false\", accent = \"false\" , font_style_name = \"2D Comment\", size = \"12\", foreground = \"[0,0 ,0]\", background = \"[255,255,255]\"), 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movablelimits = \"false\" , accent = \"false\", font_style_name = \"2D Comment\", size = \"12\", foreground = \"[0,0,0]\", background = \"[255,255,255]\"), Typesettin g:-mi(\"outer\"), Typesetting:-mo(\"⁢\", form = \"infix \", fence = \"false\", separator = \"true\", lspace = \"0em\", rspace \+ = \"verythickmathspace\", stretchy = \"false\", symmetric = \"false\", maxsize = \"infinity\", minsize = \"1\", largeop = \"false\", movable limits = \"false\", accent = \"false\", font_style_name = \"2D Comment \", size = \"12\", foreground = \"[0,0,0]\", background = \"[255,255,2 55]\"), Typesetting:-mi(\"loop\"), Typesetting:-mo(\",\", form = \"inf ix\", fence = \"false\", separator = \"true\", lspace = \"0em\", rspac e = \"verythickmathspace\", stretchy = \"false\", symmetric = \"false \", maxsize = \"infinity\", minsize = \"1\", largeop = \"false\", mova blelimits = \"false\", accent = \"false\", font_style_name = \"2D Comm ent\", size = \"12\", foreground = \"[0,0,0]\", background = 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\"12\", foreground = \"[0,0,0]\", background = \"[255,255,255]\"), Typ esetting:-mi(\"answer\"), Typesetting:-mo(\"⁢\", form = \"infix\", fence = \"false\", separator = \"false\", lspace = \"0em\" , rspace = \"0em\", stretchy = \"false\", symmetric = \"false\", maxsi ze = \"infinity\", minsize = \"1\", largeop = \"false\", movablelimits = \"false\", accent = \"false\", font_style_name = \"2D Comment\", si ze = \"12\", foreground = \"[0,0,0]\", background = \"[255,255,255]\") , Typesetting:-mi(\"to\"), Typesetting:-mo(\"⁢\", form \+ = \"infix\", fence = \"false\", separator = \"false\", lspace = \"0em \", rspace = \"0em\", stretchy = \"false\", symmetric = \"false\", max size = \"infinity\", minsize = \"1\", largeop = \"false\", movablelimi ts = \"false\", accent = \"false\", font_style_name = \"2D Comment\", \+ size = \"12\", foreground = \"[0,0,0]\", background = \"[255,255,255] \"), Typesetting:-mi(\"the\"), Typesetting:-mo(\"⁢\", f orm = \"infix\", fence = \"false\", separator = 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Typesetting:-mo(\"⁢\", form = \"infix\", fen ce = \"false\", separator = \"true\", lspace = \"0em\", rspace = \"ver ythickmathspace\", stretchy = \"false\", symmetric = \"false\", maxsiz e = \"infinity\", minsize = \"1\", largeop = \"false\", movablelimits \+ = \"false\", accent = \"false\", font_style_name = \"2D Math\", size = \"12\", foreground = \"[0,0,0]\", background = \"[255,255,255]\"), Ty pesetting:-mrow(Typesetting:-mi(\"\"), Typesetting:-mfrac(Typesetting: -mrow(Typesetting:-mi(\"\"), Typesetting:-msup(Typesetting:-mi(\"Pi\") , Typesetting:-mn(\"2\"), superscriptshift = \"0\"), Typesetting:-mo( \".\", form = \"infix\", fence = \"false\", separator = \"false\", lsp ace = \"0em\", rspace = \"0em\", stretchy = \"false\", symmetric = \"f alse\", maxsize = \"infinity\", minsize = \"1\", largeop = \"false\", \+ movablelimits = \"false\", accent = \"false\", font_style_name = \"2D \+ Math\", size = \"12\", foreground = \"[0,0,0]\", background = \"[255,2 55,255]\")), Typesetting:-mrow(Typesetting:-mn(\"6\")), linethickness \+ = \"1\", denomalign = \"center\", numalign = \"center\", bevelled = \" false\", foreground = \"[0,0,0]\", background = \"[255,255,255]\"), Ty pesetting:-mi(\"\")), Typesetting:-mi(\"\")), Typesetting:-mo(\".\", f orm = \"infix\", fence = \"false\", separator = \"false\", lspace = \" 0em\", rspace = \"0em\", stretchy = \"false\", symmetric = \"false\", \+ maxsize = \"infinity\", minsize = \"1\", largeop = \"false\", movablel imits = \"false\", accent = \"false\", font_style_name = \"2D Comment \", size = \"12\", foreground = \"[0,0,0]\", background = \"[255,255,2 55]\"));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6er-I# miGF$6#Q!F'-I&mfracGF$6*-F#6%F+-I*mverbatimGF$6#Q*)%#PiG\"\"#F'F+-F56# Q)*$\"\"'\"\"\"F'/%.linethicknessGQ\"1F'/%+denomalignGQ'centerF'/%)num alignGF@/%)bevelledGQ&falseF'/%+foregroundGQ([0,0,0]F'/%+backgroundGQ. 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" } {TEXT 204 5 "When " }{XPPEDIT 2 0 "p = 2;" "6#/%\"pG\"\"#" }{TEXT 204 13 " approximate " }{XPPEDIT 2 0 "Pi;" "6#%#PiG" }{TEXT 204 139 " to 2 decimal digits; that is, use an N big enough so that if you multiply \+ your approximation by 6 and take the square root, what you have " }} {PARA 0 "" 0 "" {TEXT 204 64 "agrees with Maple's value of Pi in the f irst two places, 3.1..." }}{PARA 0 "" 0 "" {TEXT 204 0 "" }}{PARA 0 " " 0 "" {TEXT 204 18 "3. The series for " }{XPPEDIT 18 0 "p = 4;" "6#/% \"pG\"\"%" }{TEXT 204 7 " it is " }{XPPEDIT 18 0 "Pi^4/90;" "6#*&)%#Pi G\"\"%\"\"\"\"#!*!\"\"" }{TEXT 204 32 ". Can you guess the formula for " }{XPPEDIT 18 0 "p = 6, 8, 10;" "6%/%\"pG\"\"'\"\")\"#5" }{TEXT 204 3 "? " }}{PARA 0 "" 0 "" {TEXT 204 34 "Use these formulas to approxim ate " }{XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT 204 13 ". as follows:" }} {PARA 0 "" 0 "" {TEXT 204 0 "" }}{PARA 0 "" 0 "" {TEXT 204 5 "When " } {XPPEDIT 18 0 "p = 4;" "6#/%\"pG\"\"%" }{TEXT 204 13 " approximate " } {XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT 204 57 " to 5 decimal digits- so that it gives you 3.1415. When " }{XPPEDIT 18 0 "p = 6;" "6#/%\"pG\" \"'" }{TEXT 204 13 " approximate " }{XPPEDIT 18 0 "Pi;" "6#%#PiG" } {TEXT 204 27 " to 9 decimal digits. When " }{XPPEDIT 18 0 "p = 8;" "6# /%\"pG\"\")" }{TEXT 204 13 " approximate " }{XPPEDIT 18 0 "Pi;" "6#%#P iG" }{TEXT 204 76 " to 13 decimal digits. Set the number of digits \+ you will work with to 20." }}{PARA 0 "" 0 "" {TEXT 204 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 200 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 200 0 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 204 9 "4. Let " }{XPPEDIT 18 0 "R[ N] = sum(1/n^p, n = N+1 .. infinity);" "6#/&%\"RG6#%\"NG-%$sumG6$*&\" \"\"F,)%\"nG%\"pG!\"\"/F.;,&F'F,F,F,%)infinityG" }{TEXT 204 91 ". This is the error we make when we approximate the sum of the series with t he partial sum " }{XPPEDIT 18 0 "s[N];" "6#&%\"sG6#%\"NG" }{TEXT 204 1 "." }}{PARA 0 "" 0 "" {TEXT 204 131 " Use graphs of the partial sum s and the corresponding integral to explain why for p=2 and p=4. Expl ain how the graph shows that " }}{PARA 0 "" 0 "" {TEXT 204 2 "0<" } {XPPEDIT 18 0 "R[N];" "6#&%\"RG6#%\"NG" }{TEXT 204 1 "<" }{XPPEDIT 18 0 "int(1/x^p, x = N .. infinity);" "6#-%$intG6$*&\"\"\"F')%\"xG%\"pG! \"\"/F);%\"NG%)infinityG" }{TEXT 204 2 ". " }}{PARA 0 "" 0 "" {TEXT 204 139 "5. Evaluate the improper integrals for p=2 and p=4. Estimat e the error in the approximations. That is, see how the error depends on N. ." }}}} {MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }