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their related partial sums are evaluated using " }{TEXT 435 5 "Maple" }{TEXT 436 2 ". " } {TEXT 437 0 "" }}{PARA 210 "" 0 "" {TEXT 438 96 " An elementary erro r bound theorem for partial sums based on the Integral Test is utilize d to " }{TEXT 438 0 "" }}{PARA 210 "" 0 "" {TEXT 438 46 " estimate a n infinite sum in the exercises. " }{TEXT 438 0 "" }}{PARA 208 "" 0 "" {TEXT 433 0 "" }}{PARA 211 "" 0 "" {TEXT 439 83 " Suppose that we ha ve been able to use the Integral Test to verify that the series" } {TEXT 439 0 "" }}{PARA 208 "" 0 "" {TEXT 433 0 "" }}{PARA 207 "" 0 "" {TEXT 431 37 " " }{TEXT 440 1 "S" }{TEXT 431 10 " = " }{XPPEDIT 404 0 "Sum(f(n),n=1..infinity):" "6#-%$SumG6$-%\"fG6#%\"nG/F);\"\"\"%)infinityG" }{TEXT 441 2 " " } {TEXT 431 16 " converges. " }{TEXT 431 0 "" }}{PARA 208 "" 0 "" {TEXT 433 0 "" }}{PARA 207 "" 0 "" {TEXT 431 30 " Now let \+ " }{XPPEDIT 18 0 "R[n]" "6#&%\"RG6#%\"nG" }{TEXT 431 11 " \+ = " }{XPPEDIT 392 0 "Sum(f(k),k=1..infinity)-Sum(f(k),k=1..n):" " 6#,&-%$SumG6$-%\"fG6#%\"kG/F*;\"\"\"%)infinityGF--F%6$F'/F*;F-%\"nG!\" \"" }{TEXT 441 2 " " }{TEXT 431 5 " ." }{TEXT 431 0 "" }}{PARA 208 "" 0 "" {TEXT 433 0 "" }}{PARA 207 "" 0 "" {TEXT 431 10 " Thus, \+ " }{XPPEDIT 18 0 "R[n]" "6#&%\"RG6#%\"nG" }{TEXT 431 53 " is the \+ error in approximating the infinite sum, " }{TEXT 440 1 "S" }{TEXT 431 12 ", with the n" }{TEXT 441 2 "th" }{TEXT 431 13 " partial sum." }{TEXT 431 0 "" }}{PARA 207 "" 0 "" {TEXT 431 22 " Since the function " }{TEXT 441 1 "f" }{TEXT 431 30 " is nonincreasing on [ 1 , " } {XPPEDIT 18 0 "infinity" "6#%)infinityG" }{TEXT 431 22 " ) it foll ows that" }{TEXT 431 0 "" }}{PARA 208 "" 0 "" {TEXT 433 0 "" }}{PARA 208 "" 0 "" {TEXT 433 0 "" }}{PARA 207 "" 0 "" {TEXT 431 38 " \+ " }{XPPEDIT 18 0 "R[n]" "6#&%\"RG6#%\"nG" }{TEXT 431 11 " = " }{TEXT 441 1 "f" }{TEXT 431 3 " ( " } {TEXT 441 1 "n" }{TEXT 431 11 " + 1 ) + " }{TEXT 441 1 "f" }{TEXT 431 3 " ( " }{TEXT 441 1 "n" }{TEXT 431 19 " + 2 ) + . . . " } {TEXT 442 1 "<" }{TEXT 431 4 " " }{XPPEDIT 277 0 "Int(f(x),x=n..inf inity):" "6#-%$IntG6$-%\"fG6#%\"xG/F);%\"nG%)infinityG" }{TEXT 441 2 " " }{TEXT 431 3 " ." }{TEXT 431 0 "" }}{PARA 208 "" 0 "" {TEXT 433 0 "" }}{PARA 208 "" 0 "" {TEXT 433 0 "" }}{PARA 211 "" 0 "" {TEXT 439 25 " Similarly, we have that" }{TEXT 439 0 "" }}{PARA 208 "" 0 "" {TEXT 433 0 "" }}{PARA 207 "" 0 "" {TEXT 431 38 " \+ " }{XPPEDIT 18 0 "R[n]" "6#&%\"RG6#%\"nG" }{TEXT 431 8 " = " }{TEXT 441 1 "f" }{TEXT 431 3 " ( " }{TEXT 441 1 "n" } {TEXT 431 10 " + 1 ) + " }{TEXT 441 1 "f" }{TEXT 431 3 " ( " }{TEXT 441 1 "n" }{TEXT 431 17 " + 2 ) + . . . " }{TEXT 442 1 ">" }{TEXT 431 7 " " }{XPPEDIT 275 0 "Int(f(x),x=n+1..infinity):" "6#-%$Int G6$-%\"fG6#%\"xG/F);,&%\"nG\"\"\"F.F.%)infinityG" }{TEXT 441 2 " " } {TEXT 431 5 " ." }{TEXT 431 0 "" }}{PARA 208 "" 0 "" {TEXT 433 0 "" }}{PARA 211 "" 0 "" {TEXT 439 58 " So we have geometrically verified the following theorem." }{TEXT 439 0 "" }}{PARA 211 "" 0 "" {TEXT 439 8 " " }{TEXT 439 0 "" }}{PARA 209 "" 0 "" {TEXT 443 2 " " }{TEXT 444 3 " " }{TEXT 445 149 " \+ \+ " }{TEXT 437 0 "" }}{PARA 207 "" 0 "" {TEXT 431 6 " " }{TEXT 432 40 "Remainder Estimate for the Integral Test" }{TEXT 431 0 "" }}{PARA 207 "" 0 "" {TEXT 431 15 " If " }{XPPEDIT 391 0 "Sum(f(n),n=1..infinity):" "6#-%$SumG 6$-%\"fG6#%\"nG/F);\"\"\"%)infinityG" }{TEXT 441 2 " " }{TEXT 431 43 " converges by the Integral Test and " }{XPPEDIT 18 0 "R[n]" "6 #&%\"RG6#%\"nG" }{TEXT 431 8 " = " }{TEXT 440 1 "S" }{TEXT 431 5 " - " }{XPPEDIT 18 0 "s[n]" "6#&%\"sG6#%\"nG" }{TEXT 431 11 " , \+ then" }{TEXT 431 0 "" }}{PARA 208 "" 0 "" {TEXT 433 0 "" }}{PARA 207 " " 0 "" {TEXT 431 14 " " }{TEXT 432 7 " ( 1 )" }{TEXT 431 13 " " }{TEXT 446 4 " " }{XPPEDIT 389 0 "Int(f(x),x =n+1..infinity):" "6#-%$IntG6$-%\"fG6#%\"xG/F);,&%\"nG\"\"\"F.F.%)infi nityG" }{TEXT 447 2 " " }{TEXT 446 8 " " }{TEXT 448 1 "<" } {TEXT 446 6 " " }{XPPEDIT 230 0 "R[n]" "6#&%\"RG6#%\"nG" }{TEXT 446 7 " " }{TEXT 448 1 "<" }{TEXT 446 5 " " }{XPPEDIT 388 0 "Int(f(x),x=n..infinity):" "6#-%$IntG6$-%\"fG6#%\"xG/F);%\"nG%)infinit yG" }{TEXT 447 2 " " }{TEXT 431 4 " ." }{TEXT 431 0 "" }}{PARA 208 "" 0 "" {TEXT 433 0 "" }}{PARA 208 "" 0 "" {TEXT 433 0 "" }}{PARA 207 "" 0 "" {TEXT 431 36 " Moreover, if we add " }{XPPEDIT 18 0 "s[n]" "6#&%\"sG6#%\"nG" }{TEXT 431 36 " to each side of the \+ inequality " }{TEXT 432 5 "( 1 )" }{TEXT 431 14 " we find that " } {TEXT 431 0 "" }}{PARA 208 "" 0 "" {TEXT 433 0 "" }}{PARA 207 "" 0 "" {TEXT 431 15 " " }{TEXT 432 7 " ( 2 ) " }{TEXT 431 3 " \+ " }{TEXT 446 2 " " }{XPPEDIT 226 0 "s[n]" "6#&%\"sG6#%\"nG" }{TEXT 446 12 " + " }{XPPEDIT 387 0 "Int(f(x),x=n+1..infinity):" "6# -%$IntG6$-%\"fG6#%\"xG/F);,&%\"nG\"\"\"F.F.%)infinityG" }{TEXT 447 2 " " }{TEXT 446 8 " " }{TEXT 448 1 "<" }{TEXT 446 4 " " } {TEXT 449 1 "S" }{TEXT 446 4 " " }{TEXT 448 1 "<" }{TEXT 446 6 " \+ " }{XPPEDIT 219 0 "s[n]" "6#&%\"sG6#%\"nG" }{TEXT 446 11 " + \+ " }{XPPEDIT 386 0 "Int(f(x),x=n..infinity):" "6#-%$IntG6$-%\"fG6#%\" xG/F);%\"nG%)infinityG" }{TEXT 447 2 " " }{TEXT 431 3 " ." }{TEXT 431 0 "" }}{PARA 208 "" 0 "" {TEXT 433 0 "" }}{PARA 209 "" 0 "" {TEXT 443 1 " " }{TEXT 444 5 " " }{TEXT 445 148 " \+ \+ " }{TEXT 437 0 "" }}{PARA 208 "" 0 "" {TEXT 433 0 "" }}{PARA 208 "" 0 "" {TEXT 433 0 "" }}{PARA 212 "" 0 "" {TEXT 450 11 " Example" }{TEXT 450 0 "" }} {PARA 207 "" 0 "" {TEXT 431 26 " Consider the series " }{XPPEDIT 18 0 "Sum(1/n^2,n=1..infinity):" "6#-%$SumG6$*&\"\"\"F'*$%\"nG\"\"#!\" \"/F);F'%)infinityG" }{TEXT 431 48 " . We know that this series is \+ a convergent " }{TEXT 441 1 "p" }{TEXT 431 16 " - series with " } {TEXT 441 1 "p" }{TEXT 431 6 " = 2 ." }{TEXT 431 0 "" }}{PARA 207 "" 0 "" {TEXT 431 16 " We may use " }{XPPEDIT 356 0 "s[10]" "6#&%\"sG6 #\"#5" }{TEXT 441 2 " " }{TEXT 431 23 " and the inequalities " } {TEXT 432 7 " ( 1 ) " }{TEXT 431 5 " and " }{TEXT 432 6 " ( 2 )" } {TEXT 431 32 " to estimate the infinite sum " }{TEXT 440 1 "S" } {TEXT 431 5 " = " }{XPPEDIT 18 0 "Sum(1/n^2,n=1..infinity):" "6#-%$S umG6$*&\"\"\"F'*$%\"nG\"\"#!\"\"/F);F'%)infinityG" }{TEXT 431 4 " ." }{TEXT 431 0 "" }}{PARA 207 "" 0 "" {TEXT 431 6 " In " }{TEXT 441 5 "Maple" }{TEXT 431 17 ", the command [ " }{TEXT 432 1 ">" }{TEXT 451 32 " Sum ( f ( n ) , n = a . . b )" }{TEXT 452 1 " " }{TEXT 451 2 "; " }{TEXT 452 1 " " }{TEXT 431 0 "" }}{PARA 207 "" 0 "" {TEXT 431 30 " will display the sum : " }{TEXT 432 8 "( * ) " }{TEXT 431 5 " " }{XPPEDIT 385 0 "Sum(f(n),n=a..b):" "6#-%$SumG6$-%\"fG6# %\"nG/F);%\"aG%\"bG" }{TEXT 441 2 " " }{TEXT 431 39 " in tex t form ( unevaluated )." }{TEXT 431 0 "" }}{PARA 207 "" 0 "" {TEXT 431 38 " Then the numerical value of the sum " }{TEXT 432 8 " ( * ) \+ " }{TEXT 431 30 "can be obtained by entering [" }{TEXT 432 2 " >" } {TEXT 451 14 " value ( % ) ;" }{TEXT 431 0 "" }}{PARA 208 "" 0 "" {TEXT 433 0 "" }}{PARA 207 "" 0 "" {TEXT 431 31 " Alternatively, the \+ command [" }{TEXT 432 3 " > " }{TEXT 451 31 "sum ( f ( n ) , n = a . \+ . b ) ;" }{TEXT 431 0 "" }}{PARA 207 "" 0 "" {TEXT 431 45 " will yiel d the numerical value of the sum " }{TEXT 432 5 "( * )" }{TEXT 431 41 " immediately, (provided the sum is known" }{TEXT 431 0 "" }} {PARA 207 "" 0 "" {TEXT 431 17 " in case b = + " }{XPPEDIT 18 0 "inf inity:" "6#%)infinityG" }{TEXT 431 5 " )." }{TEXT 431 0 "" }}{PARA 208 "" 0 "" {TEXT 433 0 "" }}{PARA 211 "" 0 "" {TEXT 439 51 " To solv e the given problem we proceed as follows." }{TEXT 439 0 "" }}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 430 19 "Sum(1/n^2,n=1..10);" }{MPLTEXT 1 430 0 "" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 430 16 "s[10]:=val ue(%);" }{MPLTEXT 1 430 0 "" }}}{PARA 208 "" 0 "" {TEXT 433 0 "" }} {PARA 207 "" 0 "" {TEXT 431 19 " From inequality " }{TEXT 432 5 "( 1 )" }{TEXT 431 14 " , we have " }{XPPEDIT 18 0 "R[10]" "6#&%\"RG6# \"#5" }{TEXT 431 8 " = " }{TEXT 440 1 "S" }{TEXT 431 4 " - " } {XPPEDIT 18 0 "s[10]" "6#&%\"sG6#\"#5" }{TEXT 431 28 " is bounde d below by " }{XPPEDIT 354 0 "L[10];" "6#&%\"LG6#\"#5" }{TEXT 432 2 " " }{TEXT 431 6 " and " }{TEXT 431 0 "" }}{PARA 207 "" 0 "" {TEXT 431 12 " above by " }{TEXT 432 1 " " }{XPPEDIT 352 0 "U[10];" "6#&% \"UG6#\"#5" }{TEXT 432 2 " " }{TEXT 431 9 " where :" }{TEXT 431 0 "" }}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 430 33 "L[10]:=int(1/x^2,x=11 ..infinity);" }{MPLTEXT 1 430 0 "" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 430 33 "U[10]:=int(1/x^2,x=10..infinity);" }{MPLTEXT 1 430 0 "" }}}{PARA 208 "" 0 "" {TEXT 433 0 "" }}{PARA 207 "" 0 "" {TEXT 431 20 " Using inequality " }{TEXT 432 5 "( 2 )" }{TEXT 431 41 " , w e can improve our approximation for " }{TEXT 440 1 "S" }{TEXT 431 30 " . A lower bound on the value " }{TEXT 431 0 "" }}{PARA 207 "" 0 "" {TEXT 431 14 " of the sum " }{TEXT 440 1 "S" }{TEXT 431 14 " is giv en by " }{TEXT 431 0 "" }}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 430 22 "LowerBnd:=s[10]+L[10];" }{MPLTEXT 1 430 0 "" }}}{PARA 208 "" 0 "" {TEXT 433 0 "" }}{PARA 207 "" 0 "" {TEXT 431 12 " Also from " }{TEXT 432 5 "( 2 )" }{TEXT 431 30 " , an upper bound on the sum " }{TEXT 440 1 "S" }{TEXT 431 13 " is given by" }{TEXT 431 0 "" }}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 430 22 "UpperBnd:=s[10]+U[10];" } {MPLTEXT 1 430 0 "" }}}{PARA 208 "" 0 "" {TEXT 433 0 "" }}{PARA 207 "" 0 "" {TEXT 431 55 " We may take the midpoint of the interval bounded by " }{TEXT 432 9 "LowerBnd " }{TEXT 431 6 " and " }{TEXT 432 8 "Up perBnd" }{TEXT 431 4 " as" }{TEXT 431 0 "" }}{PARA 207 "" 0 "" {TEXT 431 34 " an improved approximation for " }{TEXT 440 1 "S" }{TEXT 431 2 " ." }{TEXT 431 0 "" }}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 430 31 "Sapprox:=(LowerBnd+UpperBnd)/2;" }{MPLTEXT 1 430 0 "" }}} {PARA 208 "" 0 "" {TEXT 433 0 "" }}{PARA 207 "" 0 "" {TEXT 431 20 " \+ To ten places, " }{XPPEDIT 18 0 "Sapprox;" "6#%(SapproxG" }{TEXT 431 62 " has the value: \+ " }{TEXT 431 0 "" }}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 430 15 "eval f(Sapprox);" }{MPLTEXT 1 430 0 "" }}}{PARA 208 "" 0 "" {TEXT 433 0 "" }}{PARA 211 "" 0 "" {TEXT 439 92 " Then the maximum error in this est imate is one half the length of the interval. Therefore," }{TEXT 439 0 "" }}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 430 32 "MaxError:=(UpperB nd-LowerBnd)/2;" }{MPLTEXT 1 430 0 "" }}}{PARA 208 "" 0 "" {TEXT 433 0 "" }}{PARA 207 "" 0 "" {TEXT 431 59 " In fact, in the 1700's Leonha rd Euler verified that " }{XPPEDIT 18 0 "Sum(1/n^2,n=1..infinity) :" "6#-%$SumG6$*&\"\"\"F'*$%\"nG\"\"#!\"\"/F);F'%)infinityG" }{TEXT 431 12 " = " }{XPPEDIT 18 0 "(Pi)^2/6:" "6#*&%#PiG\"\"#\"\"'! \"\"" }{TEXT 431 8 " . " }{TEXT 431 0 "" }}{PARA 207 "" 0 "" {TEXT 431 13 " We now use " }{TEXT 441 5 "Maple" }{TEXT 431 30 " to c ompute this infinite sum." }{TEXT 431 0 "" }}{EXCHG {PARA 206 "> " 0 " " {MPLTEXT 1 430 28 "S:=sum(1/n^2,n=1..infinity);" }{MPLTEXT 1 430 0 " " }}}{PARA 208 "" 0 "" {TEXT 433 0 "" }}{PARA 207 "" 0 "" {TEXT 431 40 " Observe that the error in estimating " }{TEXT 440 1 "S" }{TEXT 431 8 " with " }{XPPEDIT 18 0 "s[10]" "6#&%\"sG6#\"#5" }{TEXT 431 15 " is given by" }{TEXT 431 0 "" }}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 430 8 "S-s[10];" }{MPLTEXT 1 430 0 "" }}}{EXCHG {PARA 206 " > " 0 "" {MPLTEXT 1 430 9 "evalf(%);" }{MPLTEXT 1 430 0 "" }}}{PARA 208 "" 0 "" {TEXT 433 0 "" }}{PARA 207 "" 0 "" {TEXT 431 43 " Clearly , this error is within the range " }{XPPEDIT 205 0 "[1/11,1/10]" "6#7 $*&\"\"\"F%\"#6!\"\"*&F%F%\"#5F'" }{TEXT 453 2 " " }{TEXT 431 39 " \+ guaranteed by the Remainder theorem." }{TEXT 431 0 "" }}{PARA 208 "" 0 "" {TEXT 433 0 "" }}{PARA 207 "" 0 "" {TEXT 431 45 " Note that the \+ actual error in estimating " }{TEXT 440 1 "S" }{TEXT 431 7 " by " }{TEXT 454 7 "Sapprox" }{TEXT 431 8 " is only" }{TEXT 431 0 "" }} {EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 430 10 "S-Sapprox;" }{MPLTEXT 1 430 0 "" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 430 9 "evalf(%);" } {MPLTEXT 1 430 0 "" }}}{PARA 207 "" 0 "" {TEXT 431 58 " A difference \+ with magnitude well below the maximum of " }{XPPEDIT 204 0 "1/220" " 6#*&\"\"\"F$\"$?#!\"\"" }{TEXT 453 2 " " }{TEXT 431 21 " guaranteed by the " }{TEXT 431 0 "" }}{PARA 211 "" 0 "" {TEXT 439 20 " Remainde r Theorem." }{TEXT 439 0 "" }}{PARA 208 "" 0 "" {TEXT 433 0 "" }} {PARA 208 "" 0 "" {TEXT 433 0 "" }}{PARA 213 "" 0 "" {TEXT 455 149 " \+ \+ \+ " }{TEXT 455 0 "" }}{PARA 208 "" 0 "" {TEXT 433 0 "" }}{PARA 207 "" 0 "" {TEXT 431 2 " " }{TEXT 432 9 "Exercises" }{TEXT 431 0 "" }}{PARA 208 "" 0 "" {TEXT 433 0 "" }}{PARA 207 "" 0 "" {TEXT 432 4 " \+ 1." }{TEXT 431 1 " " }{TEXT 432 6 " ( a )" }{TEXT 431 23 " Show that t he series " }{XPPEDIT 18 0 "Sum((ln(n)/n)^2,n=1..infinity):" "6#-%$Su mG6$*$*&-%#lnG6#%\"nG\"\"\"F+!\"\"\"\"#/F+;F,%)infinityG" }{TEXT 431 34 " is convergent by applying the " }{TEXT 441 13 "Integral Test" }{TEXT 431 2 ". " }{TEXT 431 0 "" }}{PARA 207 "" 0 "" {TEXT 431 5 " \+ " }{TEXT 432 8 " ( b ) " }{TEXT 431 15 " Observe that " }{XPPEDIT 18 0 "U[n];" "6#&%\"UG6#%\"nG" }{TEXT 431 7 " = " }{XPPEDIT 18 0 " Int((ln(t)/t)^2,t = n .. infinity);" "6#-%$IntG6$*$*&-%#lnG6#%\"tG\"\" \"F+!\"\"\"\"#/F+;%\"nG%)infinityG" }{TEXT 431 48 " is an upper bou nd on the error in estimating" }{TEXT 431 0 "" }}{PARA 207 "" 0 "" {TEXT 431 61 " \+ " }{XPPEDIT 18 0 "Sum((ln(n)/n)^2,n = 1 .. infinity)" "6#-%$SumG6 $*$*&-%#lnG6#%\"nG\"\"\"F+!\"\"\"\"#/F+;F,%)infinityG" }{TEXT 431 14 " with its " }{XPPEDIT 18 0 "n^th;" "6#)%\"nG%#thG" }{TEXT 431 15 " partial sum." }{TEXT 431 0 "" }}{PARA 208 "" 0 "" {TEXT 433 0 "" }} {PARA 211 "" 0 "" {TEXT 439 16 " " }{TEXT 439 0 "" }} {PARA 207 "" 0 "" {TEXT 431 17 " " }{TEXT 456 8 "Probl em:" }{TEXT 432 1 " " }{TEXT 431 28 " Find the smallest integer " } {TEXT 441 1 "n" }{TEXT 431 14 " for which " }{XPPEDIT 18 0 "U[n];" "6#&%\"UG6#%\"nG" }{TEXT 431 4 " " }{TEXT 442 1 "<" }{TEXT 431 8 " \+ 0.025 ." }{TEXT 431 0 "" }}{PARA 207 "" 0 "" {TEXT 431 17 " \+ " }{TEXT 447 4 "Hint" }{TEXT 446 2 " :" }{TEXT 431 0 "" }}{PARA 207 "" 0 "" {TEXT 431 15 " " }{TEXT 446 24 " Define the function " }{TEXT 447 1 "U" }{TEXT 446 3 " ( " }{TEXT 447 1 "x" } {TEXT 446 9 " ) = " }{XPPEDIT 349 0 "Int((ln(t)/t)^2,t = x .. infi nity);" "6#-%$IntG6$*$*&-%#lnG6#%\"tG\"\"\"F+!\"\"\"\"#/F+;%\"xG%)infi nityG" }{TEXT 457 2 " " }{TEXT 446 5 " in" }{TEXT 447 7 " Maple" } {TEXT 446 32 ", and then execute the following" }{TEXT 431 2 " " } {TEXT 451 2 "do" }{TEXT 431 1 " " }{TEXT 446 6 " loop:" }{TEXT 431 2 " " }{TEXT 431 0 "" }}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 430 5 "n:= 1:" }{MPLTEXT 1 430 0 "" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 430 33 "while evalf( U(n) - 0.05 > 0 ) do" }{MPLTEXT 1 430 0 "" }}{PARA 206 "> " 0 "" {MPLTEXT 1 430 7 "n:=n+1:" }{MPLTEXT 1 430 0 "" }}{PARA 206 "> " 0 "" {MPLTEXT 1 430 3 "od:" }{MPLTEXT 1 430 0 "" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 430 2 "n;" }{MPLTEXT 1 430 0 "" }}} {PARA 208 "" 0 "" {TEXT 433 0 "" }}{PARA 208 "" 0 "" {TEXT 433 0 "" }} {PARA 207 "" 0 "" {TEXT 431 7 " " }{TEXT 432 5 "( c )" }{TEXT 431 8 " Find " }{TEXT 454 7 "Sapprox" }{TEXT 431 40 " as in the text above for the value of " }{TEXT 441 1 "n" }{TEXT 431 20 " found in \+ ( b ) . " }{TEXT 431 0 "" }}{PARA 207 "" 0 "" {TEXT 431 6 " " } {TEXT 432 8 " ( d ) " }{TEXT 431 41 " Find an upper bound on the diff erence: " }{XPPEDIT 18 0 "abs(Sum((ln(n)/n)^2,n = 1 .. infinity)-S[ap prox]);" "6#-%$absG6#,&-%$SumG6$*$*&-%#lnG6#%\"nG\"\"\"F/!\"\"\"\"#/F/ ;F0%)infinityGF0&%\"SG6#%'approxGF1" }{TEXT 431 5 " ." }{TEXT 431 0 "" }}{PARA 208 "" 0 "" {TEXT 433 0 "" }}{PARA 207 "" 0 "" {TEXT 432 1 "2" }{TEXT 431 1 "." }{TEXT 432 1 " " }{TEXT 431 45 " It is not yet known whether the series " }{XPPEDIT 18 0 "Sum((1/(n^3*[sin(n)]^2 ),n=1..infinity):" "6#-%$SumG6$*&\"\"\"F'*&%\"nG\"\"$7#-%$sinG6#F)\"\" #!\"\"/F);F'%)infinityG" }{TEXT 431 33 " converges or diverg es." }{TEXT 431 0 "" }}{PARA 207 "" 0 "" {TEXT 431 7 " " }{TEXT 458 1 " " }{TEXT 459 6 "( a ) " }{TEXT 432 1 " " }{TEXT 431 22 "Try to evaluate " }{XPPEDIT 18 0 "Sum(1/(n^3*[cos(n)]^2),n = 1 .. infi nity):" "6#-%$SumG6$*&\"\"\"F'*&%\"nG\"\"$7#-%$cosG6#F)\"\"#!\"\"/F);F '%)infinityG" }{TEXT 431 20 " using " }{TEXT 441 5 "Maple " }{TEXT 431 3 " . " }{TEXT 431 0 "" }}{PARA 207 "" 0 "" {TEXT 431 15 " " }{TEXT 446 3 " " }{TEXT 460 4 "Note" }{TEXT 461 3 " : " }{TEXT 446 4 " In " }{TEXT 447 8 "Maple's " }{TEXT 446 12 " math text " }{TEXT 451 15 "(sin ( n ) ) ^2" }{TEXT 446 15 " appears as \+ " }{XPPEDIT 311 0 "(sin(n))^2" "6#*$-%$sinG6#%\"nG\"\"#" }{TEXT 451 2 " " }{TEXT 461 1 " " }{TEXT 446 3 " . " }{TEXT 431 0 "" }}{PARA 208 "" 0 "" {TEXT 433 0 "" }}{PARA 207 "" 0 "" {TEXT 431 2 " " } {TEXT 452 1 " " }{TEXT 458 5 " " }{TEXT 459 5 "( b )" }{TEXT 452 1 " " }{TEXT 431 45 " Define the ordered pair of real numbers " } {TEXT 431 0 "" }}{PARA 207 "" 0 "" {TEXT 431 50 " \+ " }{TEXT 432 4 "S [ " }{TEXT 454 1 "k" } {TEXT 431 1 " " }{TEXT 432 2 "] " }{TEXT 431 13 " = " } {TEXT 462 1 "(" }{TEXT 431 2 " " }{TEXT 441 1 "k" }{TEXT 431 10 " , " }{XPPEDIT 18 0 "Sum((1/(n^3*[sin(n)]^2),n=1..k):" "6#-%$SumG6$ *&\"\"\"F'*&%\"nG\"\"$7#-%$sinG6#F)\"\"#!\"\"/F);F'%\"kG" }{TEXT 431 8 " " }{TEXT 463 1 " " }{TEXT 462 1 ")" }{TEXT 431 0 "" }} {PARA 208 "" 0 "" {TEXT 433 0 "" }}{PARA 207 "" 0 "" {TEXT 431 23 " \+ for " }{TEXT 441 1 "k" }{TEXT 431 37 " = 1 , 2 , 3 , . . . 400 using a " }{TEXT 451 2 "do" }{TEXT 431 8 " loop ." } {TEXT 431 0 "" }}{PARA 208 "" 0 "" {TEXT 433 0 "" }}{PARA 207 "" 0 "" {TEXT 431 3 " " }{TEXT 458 3 " " }{TEXT 459 9 " ( c ) " }{TEXT 458 1 " " }{TEXT 431 29 " Plot the first 100 points " }{TEXT 432 5 " S[1[ " }{TEXT 431 1 "," }{TEXT 432 6 " S[2] " }{TEXT 431 1 "," }{TEXT 432 6 " S[3] " }{TEXT 431 2 ", " }{TEXT 432 6 ". . . " }{TEXT 431 1 ", " }{TEXT 432 8 " S[100] " }{TEXT 431 22 ", for the sequence of " } {TEXT 431 0 "" }}{PARA 211 "" 0 "" {TEXT 439 32 " pa rtial sums." }{TEXT 439 0 "" }}{PARA 211 "" 0 "" {TEXT 439 69 " \+ To do this, you may execute the following commands:" } {TEXT 439 0 "" }}{PARA 207 "" 0 "" {TEXT 431 18 " " } {TEXT 464 30 "A:=[ seq ( S[k],k=1..100 ) ] :" }{TEXT 431 0 "" }}{PARA 207 "" 0 "" {TEXT 431 18 " " }{TEXT 464 62 "plot ( A \+ , style = point , symbol = diamond, color=magenta ) ;" }{TEXT 431 0 "" }}{PARA 211 "" 0 "" {TEXT 439 59 " Do the partial s ums appear to converge ?" }{TEXT 439 0 "" }}{PARA 208 "" 0 "" {TEXT 433 0 "" }}{PARA 207 "" 0 "" {TEXT 431 7 " " }{TEXT 451 1 " " } {TEXT 459 8 " ( d ) " }{TEXT 431 61 " Plot the first 200 points for \+ the sequence of partial sums." }{TEXT 431 0 "" }}{PARA 211 "" 0 "" {TEXT 439 59 " Discuss the behavior in your own wor ds." }{TEXT 439 0 "" }}{PARA 208 "" 0 "" {TEXT 433 0 "" }}{PARA 207 "" 0 "" {TEXT 431 7 " " }{TEXT 458 2 " " }{TEXT 459 6 " ( e )" } {TEXT 458 1 " " }{TEXT 431 20 " Plot the points " }{TEXT 432 5 "S[1 ] " }{TEXT 431 2 ", " }{TEXT 432 4 "S[2]" }{TEXT 431 3 " , " }{TEXT 432 4 "S[3]" }{TEXT 431 3 " , " }{TEXT 432 5 ". . ." }{TEXT 431 3 " , \+ " }{TEXT 432 5 "S[400" }{TEXT 431 21 "] . What happens at " }{TEXT 432 7 " S[355]" }{TEXT 431 3 " ? " }{TEXT 431 0 "" }}{PARA 207 "" 0 "" {TEXT 431 44 " Calculate the number " } {XPPEDIT 203 0 "355/113" "6#*&\"$b$\"\"\"\"$8\"!\"\"" }{TEXT 453 2 " \+ " }{TEXT 431 51 " and use this result to explain the behavior at " }{TEXT 432 6 "S[355]" }{TEXT 431 2 " ." }{TEXT 431 0 "" }}{PARA 207 "" 0 "" {TEXT 431 41 " For what values of " }{TEXT 441 1 "k" }{TEXT 431 54 " would you guess that this behavior occurs a gain at " }{TEXT 432 5 "S[k] " }{TEXT 431 1 "?" }{TEXT 431 0 "" }} {EXCHG {PARA 214 "> " 0 "" {MPLTEXT 1 465 0 "" }}}{EXCHG {PARA 214 "> \+ " 0 "" {MPLTEXT 1 465 0 "" }}}{PARA 208 "" 0 "" {TEXT 433 0 "" }} {PARA 215 "" 0 "" {TEXT 466 0 "" }}{PARA 215 "" 0 "" {TEXT 466 0 "" }} {PARA 208 "" 0 "" {TEXT 433 0 "" }}{PARA 216 "" 0 "" {TEXT -1 0 "" }}} {MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }