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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT 200 28 "MATH 156 FINAL LAB PROJEC
T 2" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 204 57 "In this project you will \+
investigate the harmonic series " }{XPPEDIT 18 0 "sum(1/n, n = 1 .. in
finity);" "6#-%$sumG6$*&\"\"\"F'%\"nG!\"\"/F(;F'%)infinityG" }{TEXT 
204 4 "  . " }}{PARA 0 "" 0 "" {TEXT 204 0 "" }}{PARA 0 "" 0 "" {TEXT 
204 74 "1. Use the proof of the integral test and graphs to show the i
nequalities " }{XPPEDIT 18 0 "ln(N+1) < s[N];" "6#2-%#lnG6#,&%\"NG\"\"
\"F)F)&%\"sG6#F(" }{TEXT 204 1 "<" }{XPPEDIT 18 0 "1+ln*N;" "6#,&\"\"
\"F$*&%#lnGF$%\"NGF$F$" }{TEXT 204 2 ", " }}{PARA 0 "" 0 "" {TEXT 204 
6 "where " }{XPPEDIT 18 0 "s[N];" "6#&%\"sG6#%\"NG" }{TEXT 204 46 " ar
e the partial sums of the harmonic series. " }}{PARA 0 "" 0 "" {TEXT 
204 57 "2. Determine how many terms M you need to sum up so that " }
{XPPEDIT 18 0 "sum(1/n, n = 1 .. M);" "6#-%$sumG6$*&\"\"\"F'%\"nG!\"\"
/F(;F'%\"MG" }{TEXT 204 91 "  >50. Show that the sum of the first mill
ion terms of the harmonic series is less than 15." }}}{EXCHG {PARA 0 "
" 0 "" {TEXT 204 66 "3. Use graphs to explain why the following inequa
lities are true: " }{XPPEDIT 18 0 "ln(21/10) < 1/10+1/11+1/12;" "6#2-%
#lnG6#*&\"#@\"\"\"\"#5!\"\",(*&F)F)F*F+F)*&F)F)\"#6F+F)*&F)F)\"#7F+F)"
 }{TEXT 204 4 "+..." }{XPPEDIT 18 0 "1/20 < ln(20/9);" "6#2*&\"\"\"F%
\"#?!\"\"-%#lnG6#*&F&F%\"\"*F'" }{TEXT 204 9 "   and   " }{XPPEDIT 18 
0 "ln(201/100) < 1/100+1/101+1/102;" "6#2-%#lnG6#*&\"$,#\"\"\"\"$+\"!
\"\",(*&F)F)F*F+F)*&F)F)\"$,\"F+F)*&F)F)\"$-\"F+F)" }{TEXT 204 4 "+...
" }{XPPEDIT 18 0 "1/200 < ln(200/99);" "6#2*&\"\"\"F%\"$+#!\"\"-%#lnG6
#*&F&F%\"#**F'" }}}{EXCHG {PARA 11 "" 1 "" {TEXT 205 0 "" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT 204 38 "4. Use Maple to compute the limit     " 
}{XPPEDIT 19 1 "limit(sum(1/n, n = m .. 2*m), m = infinity);" "6#-%&li
mitG6$-%$sumG6$*&\"\"\"F*%\"nG!\"\"/F+;%\"mG*&\"\"#F*F/F*/F/%)infinity
G" }{TEXT 204 43 ". Explain your answer. Compute the limit   " }
{XPPEDIT 19 1 "limit(sum(1/n, n = 2*m .. 3*m), m = infinity);" "6#-%&l
imitG6$-%$sumG6$*&\"\"\"F*%\"nG!\"\"/F+;*&\"\"#F*%\"mGF**&\"\"$F*F1F*/
F1%)infinityG" }{TEXT 204 47 " . Explain your answer. Do you see the p
attern?" }}}}
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