{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 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }1 0 0 0 8 4 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 2" 3 4 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 8 2 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 256 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 1 1 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {PARA 4 "" 0 "" {TEXT -1 19 "Module 10 : Series" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 3 "" 0 "" {TEXT -1 27 "1004 : General Pow er Series" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{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 -1 198 "Maple contains a power package of commands created to de al with power series. Using this package, we can create arbitrary powe r series, perform operations on them, and evaluate them for convergenc e." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "restart; with(powseries):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 83 "_____________________________________ ______________________________________________" }}{PARA 4 "" 0 "" {TEXT -1 18 "A. Defining Series" }}{PARA 0 "" 0 "" {TEXT -1 83 "______ ______________________________________________________________________ _______" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 296 "In the previous module, we learned \+ how to create series for arbitrary functions using the series command. Now we are going to turn this process around, and define power series by defining what their coefficients should look like - without necess arily knowing what function if any, they represent." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 125 "To create a power seri es, we simply define a function that defines the coefficients. And the n to view the series, we use the " }{TEXT 256 8 "tpsform " }{TEXT -1 116 "command which renders the series as a \"truncated power series\". The parameter 8 tells us how many terms to display." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "truncated power series " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "powcreate( a(n) = 1/2^n); tpsform(a, x, 8);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+5%\"xG\"\"\"\"\"!#F%\"\"#F%#F%\"\"%F(#F%\"\")\"\" $#F%\"#;F*#F%\"#K\"\"&#F%\"#k\"\"'#F%\"$G\"\"\"(-%\"OG6#F%F," }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "powcreate( b(n) = 1/3^n); \+ tpsform(b, x, 8);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+5%\"xG\"\"\"\"\" !#F%\"\"$F%#F%\"\"*\"\"##F%\"#FF(#F%\"#\")\"\"%#F%\"$V#\"\"&#F%\"$H(\" \"'#F%\"%(=#\"\"(-%\"OG6#F%\"\")" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 140 "We don't need to compute everything from scratch. There are some for the more important functions' series buil t in to the powseries package." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "s := powsin( x ): tpsform( s, x, 12);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+1%\"xG\"\"\"F%#!\"\"\" \"'\"\"$#F%\"$?\"\"\"&#F'\"%S]\"\"(#F%\"'!)GO\"\"*#F'\")+o\"*R\"#6-%\" OG6#F%\"#7" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "c := powcos( \+ x ): tpsform( c, x, 12);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+1%\"xG \"\"\"\"\"!#!\"\"\"\"#F)#F%\"#C\"\"%#F(\"$?(\"\"'#F%\"&?.%\"\")#F(\"(+ )GO\"#5-%\"OG6#F%\"#7" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "e \+ := powexp( x ): tpsform( e, x, 8);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#+5%\"xG\"\"\"\"\"!F%F%#F%\"\"#F(#F%\"\"'\"\"$#F%\"#C\"\"%#F%\"$?\"\" \"&#F%\"$?(F*#F%\"%S]\"\"(-%\"OG6#F%\"\")" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 83 "_____________________________________ ______________________________________________" }}{PARA 4 "" 0 "" {TEXT -1 28 "B. Evaluating A Power Series" }}{PARA 0 "" 0 "" {TEXT -1 83 "__________________________________________________________________ _________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 168 "To evaluate a series at \+ a value, we need to follow two steps. After defining the series, we na me the truncated power series, then evaluate it at a particular using \+ the " }{TEXT 257 4 "eval" }{TEXT -1 9 " command." }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 45 "s := powsin( x ): sx := tpsform( s, x, 12); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#sxG+1%\"xG\"\"\"F'#!\"\"\"\"'\" \"$#F'\"$?\"\"\"&#F)\"%S]\"\"(#F'\"'!)GO\"\"*#F)\")+o\"*R\"#6-%\"OG6#F '\"#7" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "eval( sx, x = a + \+ b);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+1,&%\"aG\"\"\"%\"bGF&F&F&#!\" \"\"\"'\"\"$#F&\"$?\"\"\"&#F)\"%S]\"\"(#F&\"'!)GO\"\"*#F)\")+o\"*R\"#6 -%\"OG6#F&\"#7" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 32 "eval( sx, x = Pi/4); evalf(%);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#,0%#PiG#\"\"\"\"\"%*&#F&\"$%QF&*$)F$\"\"$F&F&!\" \"*&#F&\"'!)G7F&)F$\"\"&F&F&*&#F&\")g`d#)F&*$)F$\"\"(F&F&F.*&#F&\",?Z \"o7&*F&)F$\"\"*F&F&*&#F&\"0+s!R>Bu;F&*$)F$\"#6F&F&F.-%\"OG6#,$*$)F$\" #7F&#F&\");sx;F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&$\"+8y1rq!#5\"\" \"-%\"OG6#,$*$)%#PiG\"#7F'#F'\");sx;F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "evalf( sin( Pi/4));" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#$\"+5y1rq!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 83 "_______________________________ ____________________________________________________" }}{PARA 4 "" 0 " " {TEXT -1 13 "C. Operations" }}{PARA 0 "" 0 "" {TEXT -1 83 "_________ ______________________________________________________________________ ____" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 129 "One of the features which makes the po wer series package so powerful is the ability to perform arithmetic op erations with series." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "s := powadd( a,b ): tpsform( s, x, 8); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+5%\"xG\"\"#\"\"!#\"\"&\"\"'\"\"\" #\"#8\"#OF%#\"#N\"$;#\"\"$#\"#(*\"%'H\"\"\"%#\"$v#\"%wxF(#\"$$z\"&cm%F )#\"%:B\"'O*z#\"\"(-%\"OG6#F*\"\")" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "d := subtract( a, b ): tpsform(d, x,8);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+3%\"xG#\"\"\"\"\"'F&#\"\"&\"#O\"\"##\"#>\"$ ;#\"\"$#\"#l\"%'H\"\"\"%#\"$6#\"%wxF)#\"$l'\"&cm%F'#\"%f?\"'O*z#\"\"(- %\"OG6#F&\"\")" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "m := mult iply( a, b ): tpsform( m, x, 8);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# +5%\"xG\"\"\"\"\"!#\"\"&\"\"'F%#\"#>\"#O\"\"##\"#l\"$;#\"\"$#\"$6#\"%' H\"\"\"%#\"$l'\"%wxF(#\"%f?\"&cm%F)#\"%0j\"'O*z#\"\"(-%\"OG6#F%\"\")" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "q := quotient( a,b ): t psform(q,x,8);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+5%\"xG\"\"\"\"\"!#F %\"\"'F%#F%\"#7\"\"##F%\"#C\"\"$#F%\"#[\"\"%#F%\"#'*\"\"&#F%\"$#>F(#F% \"$%Q\"\"(-%\"OG6#F%\"\")" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "sm := multiply( s,m): tpsform( sm, x, 12);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+=%\"xG\"\"#\"\"!#\"\"&F%\"\"\"#\"#>\"\"*F%#\"$D$\"$;# \"\"$#\"$6#F/\"\"%#\"%bY\"%wxF(#\"%f?\"%Ke\"\"'#\"%0j\"&/6$\"\"(#\"&be *\"'3)R)\"\")#\"'v#Q'\")'px+\"F,#\"'*4v\"\"([)Q]\"#5#\"(&[bo\"*cqzi$\" #6-%\"OG6#F)\"#7" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 139 "We can use these operations to verify some common identi ties. For example, lets demonstrate the validity of cos(2*x) = cos(x)^ 2 - sin(x)^2;" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "s := powsin( x): tpsform(s,x,12);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#+1%\"xG\"\"\"F%#!\"\"\"\"'\"\"$#F%\"$?\"\"\"&#F' \"%S]\"\"(#F%\"'!)GO\"\"*#F'\")+o\"*R\"#6-%\"OG6#F%\"#7" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "c := powcos(x): tpsform(c, x,12); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+1%\"xG\"\"\"\"\"!#!\"\"\"\"#F)#F% \"#C\"\"%#F(\"$?(\"\"'#F%\"&?.%\"\")#F(\"(+)GO\"#5-%\"OG6#F%\"#7" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "s2 := multiply( s,s): tpsf orm( s2, x, 12);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+/%\"xG\"\"\"\"\"# #!\"\"\"\"$\"\"%#F&\"#X\"\"'#F(\"$:$\"\")#F&\"&vT\"\"#5-%\"OG6#F%\"#7 " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "c2 := multiply( c,c): \+ tpsform( c2, x, 12);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+1%\"xG\"\"\" \"\"!!\"\"\"\"##F%\"\"$\"\"%#!\"#\"#X\"\"'#F%\"$:$\"\")#F-\"&vT\"\"#5- %\"OG6#F%\"#7" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "c2s2 := su btract( c2, s2): tpsform( c2s2, x, 12);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+1%\"xG\"\"\"\"\"!!\"#\"\"##F(\"\"$\"\"%#!\"%\"#X\"\"'# F(\"$:$\"\")#F-\"&vT\"\"#5-%\"OG6#F%\"#7" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "c2x := powcos( 2*x): tpsform( c2x, x, 12);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#+1%\"xG\"\"\"\"\"!!\"#\"\"##F(\"\"$\" \"%#!\"%\"#X\"\"'#F(\"$:$\"\")#F-\"&vT\"\"#5-%\"OG6#F%\"#7" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "difference := subtract( c2s2, c2x): tpsform( difference, x, 12);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+%% \"xG-%\"OG6#\"\"\"\"#7" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 74 "Here is a famous demonstration of the fact that e^ ix = cos(x) + i*sin(x)" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "c := powcos( x ): tpsform(s, x, 1 2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+1%\"xG\"\"\"F%#!\"\"\"\"'\"\"$ #F%\"$?\"\"\"&#F'\"%S]\"\"(#F%\"'!)GO\"\"*#F'\")+o\"*R\"#6-%\"OG6#F%\" #7" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "si := multconst( s, I ): tpsform(si, x, 12);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+1%\"xG^# \"\"\"F&^##!\"\"\"\"'\"\"$^##F&\"$?\"\"\"&^##F)\"%S]\"\"(^##F&\"'!)GO \"\"*^##F)\")+o\"*R\"#6-%\"OG6#F&\"#7" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "cis := powadd( c, si): tpsform( cis, x, 12);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#+=%\"xG\"\"\"\"\"!^#F%F%#!\"\"\"\"#F*^ ##F)\"\"'\"\"$#F%\"#C\"\"%^##F%\"$?\"\"\"&#F)\"$?(F-^##F)\"%S]\"\"(#F% \"&?.%\"\")^##F%\"'!)GO\"\"*#F)\"(+)GO\"#5^##F)\")+o\"*R\"#6-%\"OG6#F% \"#7" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "e := powexp( I*x): \+ tpsform(e, x, 12);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+=%\"xG\"\"\" \"\"!^#F%F%#!\"\"\"\"#F*^##F)\"\"'\"\"$#F%\"#C\"\"%^##F%\"$?\"\"\"&#F) \"$?(F-^##F)\"%S]\"\"(#F%\"&?.%\"\")^##F%\"'!)GO\"\"*#F)\"(+)GO\"#5^## F)\")+o\"*R\"#6-%\"OG6#F%\"#7" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "difference := subtract( cis, e): tpsform( difference, x, 12); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+%%\"xG-%\"OG6#\"\"\"\"#7" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 178 "Another \+ operation is the composition of tow series. This is similar to taking \+ the composition of two functions f(g(x)). For example, lets compose th e series for e^x with sin(x)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "es := compose( e,s ): tps form(es, x, 8);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+5%\"xG\"\"\"\"\"!^ #F%F%#!\"\"\"\"#F*^##F)\"\"$F-#\"\"&\"#C\"\"%^##F%\"#5F/#!#P\"$?(\"\"' ^##!\")\"$:$\"\"(-%\"OG6#F%\"\")" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "series( exp( sin(x)), x, 8);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+3%\"xG\"\"\"\"\"!F%F%#F%\"\"#F(#!\"\"\"\")\"\"%#F*\"#: \"\"&#F*\"$S#\"\"'#F%\"#!*\"\"(-%\"OG6#F%F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 17 "Using the normal " }{TEXT 258 6 "series" }{TEXT -1 48 " command, we verify that we get the same resu lt." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 83 "_________________________________________ __________________________________________" }}{PARA 4 "" 0 "" {TEXT -1 28 "D. The Radius of Convergence" }}{PARA 0 "" 0 "" {TEXT -1 83 "__ ______________________________________________________________________ ___________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 217 "In any use of power series, we must consider the interval of convergence, outside of which, the powe r series has no relevance. If we define the coefficients of a power se ries, how do we know the radius of convergence?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 64 "First we re-initialize th e variable c, and we define the series." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "c := 'c';" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%\"cGF$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "powcreate( c(n) = 11*(n^2)/(2^n)): tpsform( c, x, 10);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#+7%\"xG#\"#6\"\"#\"\"\"F&F'#\"#**\"\") \"\"$F&\"\"%#\"$v#\"#K\"\"&#F*\"#;\"\"'#\"$R&\"$G\"\"\"(#F&F-F+#\"$\"* )\"$7&\"\"*-%\"OG6#F(\"#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 248 "Then we express the terms c^n+1, and c^n in a st andard way. This step, which might be difficult to understand , is nec essary because of the way Maple stores the coefficients. what we are d oing is substitute n for the interal parameter used by Maple" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "su bs(_k = n, c(_k)); subs(_k = n + 1, c(_k) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&*$)%\"nG\"\"#\"\"\"F))F(F'!\"\"\"#6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&*$),&%\"nG\"\"\"F)F)\"\"#F)F))F*F'!\"\"\"#6" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 140 "Next w e take the ratio, the absolute value of its simplification, the limit \+ as n approaches infinity, and finally the radius of convergence." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "% / %%;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&*&),&%\"nG\"\"\"F(F(\" \"#F()F)F'F(F(*&)F)F&F()F'F)F(!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "abs( simplify( %));" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#,$-%$absG6#*&*$),&%\"nG\"\"\"F,F,\"\"#F,F,*$)F+F-F,!\"\"#F,F-" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "ratio := abs( limit( %, n = \+ infinity));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&ratioG#\"\"\"\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "RadConv := 1/ ratio;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%(RadConvG\"\"#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 58 "Here is another example. \+ The series is entirely different." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "powcreate( t(n) = ( 3^n )/n!): tpsform(t,x,10);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#+9%\"xG\"\"\"\"\"!\"\"$F%#\"\"*\"\"#F* F(F'#\"#F\"\")\"\"%#\"#\")\"#S\"\"&#F0\"#!)\"\"'#\"$V#\"$g&\"\"(#\"$H( \"%![%F-#F7F " 0 "" {MPLTEXT 1 0 45 "subs(_k = n, t(_k)); subs(_k = n + 1, t(_k));" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#*&)\"\"$%\"nG\"\"\"-%*factorialG6#F&!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&)\"\"$,&%\"nG\"\"\"F(F(F(-%*factorialG6#F& !\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "%/%%;" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#*&*&)\"\"$,&%\"nG\"\"\"F)F)F)-%*factorialG6#F(F) F)*&-F+6#F'F))F&F(F)!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "abs(simplify( %));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\"\"\"F% -%$absG6#,&%\"nGF%F%F%!\"\"\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "ratio := abs( limit( %, n = infinity));" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%&ratioG\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "RadConv := limit( 1/r, r = ratio);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%(RadConvG%*undefinedG" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 75 "We use a limit to define the radiu s of convergence since the ratio is zero." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{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 }