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terms of the sequence " }{XPPEDIT 18 0 "a(n) = 1/n;" "6#/-%\"a G6#%\"nG*&\"\"\"F)F'!\"\"" }{TEXT 264 0 "" }}{PARA 203 "" 0 "" {TEXT 262 0 "" }}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 265 8 "restart;" } {MPLTEXT 1 265 0 "" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 265 17 "a :=array(1..100);" }{MPLTEXT 1 265 0 "" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 265 36 "for n from 1 to 100 do a[n]:=1/n:od:" }{MPLTEXT 1 265 0 "" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 265 43 "plot(\{seq([ i,a[i]],i=1..100)\},style=point);" }{MPLTEXT 1 265 0 "" }}}{EXCHG {PARA 203 "" 0 "" {TEXT 262 0 "" }}{PARA 207 "" 0 "" {TEXT 266 57 "We \+ can observe that the terms of the sequence approach 0." }{TEXT 266 0 " " }}{PARA 208 "" 0 "" {TEXT 267 35 "The limit command can also be used ." }{TEXT 267 0 "" }}{PARA 203 "" 0 "" {TEXT 262 0 "" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 265 22 "limit(1/i,i=infinity);" }{MPLTEXT 1 265 0 "" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 265 0 "" }}}{PARA 203 "" 0 "" {TEXT 262 0 "" }}{SECT 0 {PARA 209 "" 0 "" {TEXT 268 48 "E xample 2: Plot the partial sums of the series " }{XPPEDIT 18 0 "sum(1 /n,n = 1 .. i);" "6#-%$sumG6$*&\"\"\"F'%\"nG!\"\"/F(;F'%\"iG" }{TEXT 269 0 "" }}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 265 8 "restart;" } {MPLTEXT 1 265 0 "" }}}{EXCHG {PARA 203 "" 0 "" {TEXT 262 0 "" }} {PARA 210 "" 0 "" {TEXT 270 21 "Choose a value for i." }{TEXT 270 0 "" }}{PARA 203 "" 0 "" {TEXT 262 0 "" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 265 7 "i:=300:" }{MPLTEXT 1 265 0 "" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 265 30 "a:=array(1..i);s:=array(1..i);" }{MPLTEXT 1 265 0 "" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 265 34 "for n from 1 to i do a[n]:=1/n:od:" }{MPLTEXT 1 265 0 "" }}}{EXCHG {PARA 206 "> \+ " 0 "" {MPLTEXT 1 265 49 "for j from 1 to i do s[j]:=sum(a[n1],n1=1..j ):od:" }{MPLTEXT 1 265 0 "" }}{PARA 206 "> " 0 "" {MPLTEXT 1 265 41 "p lot(\{seq([j,s[j]],j=1..i)\},style=point);" }{MPLTEXT 1 265 0 "" }}} {EXCHG {PARA 203 "" 0 "" {TEXT 262 0 "" }}{PARA 211 "" 0 "" {TEXT 271 102 "The sequence of partial sums is not approaching a limit. This ca n be seen by evaluating some of them." }{TEXT 271 0 "" }}{PARA 203 "" 0 "" {TEXT 262 0 "" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 265 70 "e valf(s[100]);evalf(s[150]);evalf(s[200]);evalf(s[250]);evalf(s[300]);" }{MPLTEXT 1 265 0 "" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 265 0 " " }}}{EXCHG {PARA 203 "" 0 "" {TEXT 262 0 "" }}{PARA 212 "" 0 "" {TEXT 272 135 "This series has been proven to diverge by other methods . The above is not a proof, but can serve to illustrate the idea of d ivergence." }{TEXT 272 0 "" }}{PARA 203 "" 0 "" {TEXT 262 0 "" }}}} {PARA 203 "" 0 "" {TEXT 262 0 "" }}{SECT 0 {PARA 209 "" 0 "" {TEXT 268 51 "Example 3: Investigate the behavior of the series " } {XPPEDIT 18 0 "sum(1/(n^2),n = 1 .. infinity);" "6#-%$sumG6$*&\"\"\"F' *$%\"nG\"\"#!\"\"/F);F'%)infinityG" }{TEXT 269 2 " ." }{TEXT 269 0 "" }}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 265 8 "restart;" }{MPLTEXT 1 265 0 "" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 265 17 "a:=array(1.. 100);" }{MPLTEXT 1 265 0 "" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 265 38 "for n from 1 to 100 do a[n]:=1/n^2:od:" }{MPLTEXT 1 265 0 "" } }}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 265 64 "plot(\{seq([i,a[i]],i= 1..100)\},view=[0..100,0..0.3],style=point);" }{MPLTEXT 1 265 0 "" }}} {EXCHG {PARA 203 "" 0 "" {TEXT 262 0 "" }}{PARA 213 "" 0 "" {TEXT 273 57 "We can observe that the terms of the sequence approach 0." }{TEXT 273 0 "" }}{PARA 214 "" 0 "" {TEXT 274 35 "The limit command can also \+ be used." }{TEXT 274 0 "" }}{PARA 203 "" 0 "" {TEXT 262 0 "" }}} {EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 265 22 "limit(1/i,i=infinity);" }{MPLTEXT 1 265 0 "" }}}{EXCHG {PARA 203 "" 0 "" {TEXT 262 0 "" }} {PARA 215 "" 0 "" {TEXT 275 88 "The limit of the nth term is 0, so the sequence converges and the series could converge." }{TEXT 275 0 "" }} {PARA 203 "" 0 "" {TEXT 262 0 "" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 265 8 "restart;" }{MPLTEXT 1 265 0 "" }}}{EXCHG {PARA 203 " " 0 "" {TEXT 262 0 "" }}{PARA 216 "" 0 "" {TEXT 276 21 "Choose a value for i." }{TEXT 276 0 "" }}{PARA 203 "" 0 "" {TEXT 262 0 "" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 265 7 "i:=300:" }{MPLTEXT 1 265 0 "" }} }{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 265 30 "a:=array(1..i);s:=array (1..i);" }{MPLTEXT 1 265 0 "" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 265 36 "for n from 1 to i do a[n]:=1/n^2:od:" }{MPLTEXT 1 265 0 "" } }}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 265 49 "for j from 1 to i do s [j]:=sum(a[n1],n1=1..j):od:" }{MPLTEXT 1 265 0 "" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 265 41 "plot(\{seq([j,s[j]],j=1..i)\},style=point );" }{MPLTEXT 1 265 0 "" }}}{EXCHG {PARA 203 "" 0 "" {TEXT 262 0 "" }} {PARA 217 "" 0 "" {TEXT 277 131 "Compare this graph with the one in ex ample 2 and observe that it approaches a limit much more quickly and c an be shown to converge." }{TEXT 277 0 "" }}{PARA 203 "" 0 "" {TEXT 262 0 "" }}{PARA 218 "" 0 "" {TEXT 278 47 "Find some partial sums to i llustrate this fact." }{TEXT 278 0 "" }}{PARA 203 "" 0 "" {TEXT 262 0 "" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 265 69 "evalf(s[50]);evalf (s[100]);evalf(s[150]);evalf(s[200]);evalf(s[250]);" }{MPLTEXT 1 265 0 "" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 265 0 "" }}}{EXCHG {PARA 203 "" 0 "" {TEXT 262 0 "" }}{PARA 219 "" 0 "" {TEXT 279 51 "If \+ a series converges, Maple can often find the sum" }{TEXT 279 0 "" }} {PARA 203 "" 0 "" {TEXT 262 0 "" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 265 34 "n:='n':#This command initializes n" }{MPLTEXT 1 265 0 "" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 265 25 "sum(1/n^2,n= 1..infinity);" }{MPLTEXT 1 265 0 "" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 265 0 "" }}}}}}{SECT 0 {PARA 204 "" 0 "" {TEXT 263 10 "Exer cises " }{TEXT 263 0 "" }}{SECT 0 {PARA 205 "" 0 "" {TEXT 264 107 "Exe rcise 1: Plot the sequence of terms and the sequence of partial sums \+ as above and decide if the series " }{XPPEDIT 18 0 "sum(1/(n^3+5*n),n \+ = 1 .. infinity);" "6#-%$sumG6$*&\"\"\"F',&*$%\"nG\"\"$F'*&\"\"&F'F*F' F'!\"\"/F*;F'%)infinityG" }{TEXT 264 55 " converges or diverges. If i t converges, find the sum." }{TEXT 264 0 "" }}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 265 0 "" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 265 0 "" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 265 0 "" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 265 0 "" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 265 0 "" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 265 0 "" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 265 0 "" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 265 0 "" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 265 0 "" }}}}{PARA 203 "" 0 "" {TEXT 262 0 "" }}{SECT 0 {PARA 205 "" 0 "" {TEXT 264 107 "Exercise 2: Plot the sequence of terms and the s equence of partial sums as above and decide if the series " }{XPPEDIT 18 0 "sum(ln(n)/n,n = 2 .. infinity);" "6#-%$sumG6$*&-%#lnG6#%\"nG\"\" \"F*!\"\"/F*;\"\"#%)infinityG" }{TEXT 264 55 " converges or diverges. \+ If it converges, find the sum." }{TEXT 264 0 "" }}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 265 0 "" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 265 0 "" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 265 0 "" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 265 0 "" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 265 0 "" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 265 0 "" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 265 0 "" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 265 0 "" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 265 0 "" }}}}}{PARA 203 "" 0 "" {TEXT 262 0 "" }}{PARA 220 "" 0 "" {TEXT 280 14 "The Ratio Test" }{TEXT 280 0 "" }}{PARA 203 "" 0 "" {TEXT 262 0 "" }}{SECT 0 {PARA 204 "" 0 "" {TEXT 263 108 "The ratio te st can be used to determine if a series converges. If one converges, \+ the sum can then be found." }{TEXT 263 0 "" }}{PARA 203 "" 0 "" {TEXT 262 0 "" }}{PARA 203 "" 0 "" {TEXT 261 13 "For a series " }{XPPEDIT 255 0 "sum(a[n],n = 1 .. infinity);" "6#-%$sumG6$&%\"aG6#%\"nG/F);\"\" \"%)infinityG" }{TEXT 281 1 " " }{TEXT 262 2 ", " }{TEXT 261 3 "if " } {XPPEDIT 253 0 "limit(a[n+1]/a[n],n = infinity) = L;" "6#/-%&limitG6$* &&%\"aG6#,&%\"nG\"\"\"F-F-F-&F)6#F,!\"\"/F,%)infinityG%\"LG" }{TEXT 281 1 " " }{TEXT 262 2 ", " }{TEXT 261 82 "then the series converges i f L < 1, diverges if L > 1, and can do either if L = 1." }{TEXT 262 0 "" }}}{PARA 203 "" 0 "" {TEXT 262 0 "" }}{SECT 0 {PARA 204 "" 0 "" {TEXT 263 10 "Examples I" }{TEXT 263 0 "" }}{PARA 203 "" 0 "" {TEXT 261 25 "Example 1: Determine if " }{XPPEDIT 18 0 "sum(exp(n)/n!,n = 0 .. infinity);" "6#-%$sumG6$*&-%$expG6#%\"nG\"\"\"-%*factorialGF)!\"\" /F*;\"\"!%)infinityG" }{TEXT 262 1 " " }{TEXT 261 37 "converges. If i t does, find the sum." }{TEXT 262 0 "" }}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 265 8 "restart;" }{MPLTEXT 1 265 0 "" }}}{EXCHG {PARA 206 " > " 0 "" {MPLTEXT 1 265 48 "Limit((exp(n+1)/(n+1)!)/(exp(n)/n!),n=infi nity);" }{MPLTEXT 1 265 0 "" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 265 9 "value(%);" }{MPLTEXT 1 265 0 "" }}}{EXCHG {PARA 203 "" 0 "" {TEXT 262 0 "" }}{PARA 221 "" 0 "" {TEXT 282 34 "Since 0 < 1, the seri es converges." }{TEXT 282 0 "" }}{PARA 203 "" 0 "" {TEXT 262 0 "" }}} {EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 265 29 "Sum(exp(n)/n!,n=0..infin ity);" }{MPLTEXT 1 265 0 "" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 265 9 "evalf(%);" }{MPLTEXT 1 265 0 "" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 265 0 "" }}}{PARA 203 "" 0 "" {TEXT 262 0 "" }}{PARA 203 "" 0 "" {TEXT 261 25 "Example 2: Determine if " }{XPPEDIT 246 0 "sum(n^ 3/exp(n),n = 0 .. infinity);" "6#-%$sumG6$*&%\"nG\"\"$-%$expG6#F'!\"\" /F';\"\"!%)infinityG" }{TEXT 281 1 " " }{TEXT 261 37 "converges. If i t does, find the sum." }{TEXT 262 0 "" }}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 265 8 "restart;" }{MPLTEXT 1 265 0 "" }}}{EXCHG {PARA 206 " > " 0 "" {MPLTEXT 1 265 50 "Limit(((n+1)^3/exp(n+1))/(n^3/exp(n)),n=in finity);" }{MPLTEXT 1 265 0 "" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 265 9 "value(%);" }{MPLTEXT 1 265 0 "" }}}{EXCHG {PARA 203 "" 0 "" {TEXT 262 0 "" }}}{EXCHG {PARA 203 "" 0 "" {TEXT 262 0 "" }}{PARA 222 "" 0 "" {TEXT 283 57 "The limit is less than 1, therefore the series c onverges." }{TEXT 283 0 "" }}{PARA 203 "" 0 "" {TEXT 262 0 "" }}} {EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 265 30 "Sum(n^3/exp(n),n=0..infi nity);" }{MPLTEXT 1 265 0 "" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 265 16 "evalf(value(%));" }{MPLTEXT 1 265 0 "" }}}{EXCHG {PARA 206 "> \+ " 0 "" {MPLTEXT 1 265 0 "" }}}{PARA 203 "" 0 "" {TEXT 262 0 "" }} {PARA 203 "" 0 "" {TEXT 261 36 "Example 3: Determine if the series " }{XPPEDIT 232 0 "sum(2^n/(n^4),n = 1 .. infinity);" "6#-%$sumG6$*&)\" \"#%\"nG\"\"\"*$F)\"\"%!\"\"/F);F*%)infinityG" }{TEXT 262 1 " " } {TEXT 261 37 "converges. If it does, find the sum." }{TEXT 262 0 "" } }{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 265 8 "restart;" }{MPLTEXT 1 265 0 "" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 265 46 "Limit((2^(n+ 1)/(n+1)^4)/(2^n/n^4),n=infinity);" }{MPLTEXT 1 265 0 "" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 265 9 "value(%);" }{MPLTEXT 1 265 0 "" }}}{EXCHG {PARA 203 "" 0 "" {TEXT 262 0 "" }}{PARA 223 "" 0 "" {TEXT 284 59 "The limit is greater than 1, therefore the series diverges." } {TEXT 284 0 "" }}{PARA 203 "" 0 "" {TEXT 262 0 "" }}}}{PARA 203 "" 0 " " {TEXT 262 0 "" }}{PARA 203 "" 0 "" {TEXT 262 0 "" }}{SECT 0 {PARA 204 "" 0 "" {TEXT 263 10 "Exercises " }{TEXT 263 0 "" }}{PARA 203 "" 0 "" {TEXT 261 59 "Exercise 3: Use the ratio test to determine if the \+ series " }{XPPEDIT 228 0 "sum(n*2^n/(3^n),n = 1 .. infinity);" "6#-%$s umG6$*(%\"nG\"\"\")\"\"#F'F()\"\"$F'!\"\"/F';F(%)infinityG" }{TEXT 262 1 " " }{TEXT 261 38 " converges. If it does, find the sum." } {TEXT 262 0 "" }}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 265 8 "restart; " }{MPLTEXT 1 265 0 "" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 265 0 "" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 265 0 "" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 265 0 "" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 265 0 "" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 265 0 "" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 265 0 "" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 265 0 "" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 265 0 "" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 265 0 "" }}} {EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 265 0 "" }}}{EXCHG {PARA 206 "> \+ " 0 "" {MPLTEXT 1 265 0 "" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 265 0 "" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 265 0 "" }}}{PARA 203 "" 0 "" {TEXT 262 0 "" }}{PARA 203 "" 0 "" {TEXT 261 59 "Exercise \+ 4: Use the ratio test to determine if the series " }{XPPEDIT 226 0 "s um(n/(2^n),n = 1 .. infinity);" "6#-%$sumG6$*&%\"nG\"\"\")\"\"#F'!\"\" /F';F(%)infinityG" }{TEXT 262 2 " " }{TEXT 261 42 "converges. If it \+ converges, find the sum." }{TEXT 262 0 "" }}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 265 8 "restart;" }{MPLTEXT 1 265 0 "" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 265 0 "" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 265 0 "" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 265 0 "" }}} {EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 265 0 "" }}}{EXCHG {PARA 206 "> \+ " 0 "" {MPLTEXT 1 265 0 "" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 265 0 "" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 265 0 "" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 265 0 "" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 265 0 "" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 265 0 "" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 265 0 "" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 265 0 "" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 265 0 "" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 265 0 "" }}} {EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 265 0 "" }}}}{PARA 203 "" 0 "" {TEXT 262 0 "" }}{PARA 203 "" 0 "" {TEXT 262 0 "" }}{SECT 0 {PARA 204 "" 0 "" {TEXT 263 9 "Examples " }{TEXT 263 0 "" }}{PARA 203 "" 0 "" {TEXT 262 0 "" }}{PARA 224 "" 0 "" {TEXT 285 86 "The ratio test can al so be used to find the interval of convergence of a power series." } {TEXT 285 0 "" }}{PARA 203 "" 0 "" {TEXT 262 0 "" }}{PARA 203 "" 0 "" {TEXT 261 59 "Example 1: Find the interval of convergence of the seri es " }{XPPEDIT 215 0 "sum((x-2)^n,n = 0 .. infinity);" "6#-%$sumG6$),& %\"xG\"\"\"\"\"#!\"\"%\"nG/F,;\"\"!%)infinityG" }{TEXT 262 2 " ." } {TEXT 262 0 "" }}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 265 8 "restart; " }{MPLTEXT 1 265 0 "" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 265 18 "f:=(n,x)->(x-2)^n;" }{MPLTEXT 1 265 0 "" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 265 37 "a:=limit(f(n+1,x)/f(n,x),n=infinity);" } {MPLTEXT 1 265 0 "" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 265 16 "s olve(abs(a)<1);" }{MPLTEXT 1 265 0 "" }}}{EXCHG {PARA 203 "" 0 "" {TEXT 262 0 "" }}{PARA 225 "" 0 "" {TEXT 286 49 "The endpoints of the \+ interval must now be tested." }{TEXT 286 0 "" }}{PARA 203 "" 0 "" {TEXT 262 0 "" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 265 34 "Test_1 :=Sum(f(n,1),n=0..infinity);" }{MPLTEXT 1 265 0 "" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 265 9 "value(%);" }{MPLTEXT 1 265 0 "" }}} {EXCHG {PARA 203 "" 0 "" {TEXT 262 0 "" }}{PARA 226 "" 0 "" {TEXT 287 27 "The series above diverges.." }{TEXT 287 0 "" }}}{EXCHG {PARA 206 " > " 0 "" {MPLTEXT 1 265 34 "Test_2:=Sum(f(n,3),n=0..infinity);" } {MPLTEXT 1 265 0 "" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 265 9 "va lue(%);" }{MPLTEXT 1 265 0 "" }}}{EXCHG {PARA 227 "" 0 "" {TEXT 288 41 "Therefore, neither enedpoint is included." }{TEXT 288 0 "" }} {PARA 203 "" 0 "" {TEXT 262 0 "" }}{PARA 228 "" 0 "" {TEXT 289 41 "The interval of convergence is 1 < x < 3." }{TEXT 289 0 "" }}}{PARA 203 " " 0 "" {TEXT 262 0 "" }}{PARA 203 "" 0 "" {TEXT 261 59 "Example 2: Fi nd the interval of convergence of the series " }{XPPEDIT 207 0 "sum(x^ n/(n*4^n),n = 1 .. infinity);" "6#-%$sumG6$*&)%\"xG%\"nG\"\"\"*&F)F*) \"\"%F)F*!\"\"/F);F*%)infinityG" }{TEXT 262 3 " ." }{TEXT 262 0 "" }} {EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 265 8 "restart;" }{MPLTEXT 1 265 0 "" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 265 22 "f:=(n,x)->x^ n/(n*4^n);" }{MPLTEXT 1 265 0 "" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 265 37 "a:=limit(f(n+1,x)/f(n,x),n=infinity);" }{MPLTEXT 1 265 0 "" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 265 16 "solve(abs(a) <1);" }{MPLTEXT 1 265 0 "" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 265 35 "Test_1:=Sum(f(n,-4),n=1..infinity);" }{MPLTEXT 1 265 0 "" }}} {EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 265 9 "value(%);" }{MPLTEXT 1 265 0 "" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 265 33 "Test_2:Sum(f (n,4),n=1..infinity);" }{MPLTEXT 1 265 0 "" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 265 9 "value(%);" }{MPLTEXT 1 265 0 "" }}}{EXCHG {PARA 203 "" 0 "" {TEXT 262 0 "" }}{PARA 229 "" 0 "" {TEXT 290 49 "Thus, the interval of convergence is \2264 <= x < 4." }{TEXT 290 0 "" }}{PARA 203 "" 0 "" {TEXT 262 0 "" }}}}{PARA 203 "" 0 "" {TEXT 262 0 "" }} {PARA 203 "" 0 "" {TEXT 262 0 "" }}{SECT 0 {PARA 204 "" 0 "" {TEXT 263 12 "Exercises II" }{TEXT 263 0 "" }}{PARA 203 "" 0 "" {TEXT 261 60 "Exercise 5: Find the interval of convergence of the series " } {XPPEDIT 204 0 "sum((-1)^n*(x-1)^n/(n*3^n),n = 1 .. infinity);" "6#-%$ sumG6$*(),$\"\"\"!\"\"%\"nGF)),&%\"xGF)F)F*F+F)*&F+F))\"\"$F+F)F*/F+;F )%)infinityG" }{TEXT 262 2 " ." }{TEXT 262 0 "" }}{EXCHG {PARA 206 "> \+ " 0 "" {MPLTEXT 1 265 8 "restart;" }{MPLTEXT 1 265 0 "" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 265 0 "" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 265 0 "" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 265 0 "" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 265 0 "" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 265 0 "" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 265 0 "" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 265 0 "" }}} {EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 265 0 "" }}}{EXCHG {PARA 206 "> \+ " 0 "" {MPLTEXT 1 265 0 "" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 265 0 "" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 265 0 "" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 265 0 "" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 265 0 "" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 265 0 "" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 265 0 "" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 265 0 "" }}}{PARA 203 "" 0 "" {TEXT 262 0 "" }} {PARA 203 "" 0 "" {TEXT 261 65 "Exercise 6: Determine the interval of convergence of the series " }{XPPEDIT 202 0 "sum(2^n*(x+2)^n/(n^2),n \+ = 1 .. infinity);" "6#-%$sumG6$*()\"\"#%\"nG\"\"\"),&%\"xGF*F(F*F)F**$ F)F(!\"\"/F);F*%)infinityG" }{TEXT 281 1 " " }{TEXT 262 1 "." }{TEXT 262 0 "" }}{PARA 203 "" 0 "" {TEXT 262 0 "" }}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 265 8 "restart;" }{MPLTEXT 1 265 0 "" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 265 0 "" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 265 0 "" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 265 0 "" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 265 0 "" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 265 0 "" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 265 0 "" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 265 0 "" }}} {EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 265 0 "" }}}{EXCHG {PARA 206 "> \+ " 0 "" {MPLTEXT 1 265 0 "" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 265 0 "" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 265 0 "" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 265 0 "" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 265 0 "" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 265 0 "" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 265 0 "" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 265 0 "" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 265 0 "" }}}}{PARA 203 "" 0 "" {TEXT 262 0 "" }}{PARA 203 "" 0 "" {TEXT 262 0 "" }}{PARA 203 "" 0 "" {TEXT 262 0 "" }}{PARA 230 "" 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 }