{VERSION 6 0 "IBM INTEL NT" "6.0" } {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 Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 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 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }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 0 }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 0 }0 0 0 -1 8 2 0 0 0 0 0 0 -1 0 }{PSTYLE "Headi ng 3" 4 5 1 {CSTYLE "" -1 -1 "" 1 12 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }0 0 0 -1 0 0 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 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 16 "Integrace - rady" }}} {SECT 0 {PARA 5 "" 0 "" {TEXT -1 6 "Zad\341n\355" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 76 "Rozvinut\355m vhodn\351 funkce v mocninnou \370adu j est n\341m p\370ev\351sti v \370adu integr\341l:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "Int(exp(-x)*cos(sqrt(x)),x=0..infinity);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&-%$expG6#,$%\"xG!\"\"\"\"\" -%$cosG6#*$F+#F-\"\"#F-/F+;\"\"!%)infinityG" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 50 "Ov\354\370en\355 p\370edpoklad\371 v\354ty o z\341m \354n\354 sumy a integr\341lu" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "V hodnou funkc\355 k rozveden\355 je zde " }{XPPEDIT 18 0 "cos(sqrt(x)) " "6#-%$cosG6#-%%sqrtG6#%\"xG" }{TEXT -1 1 ":" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "cos(x)=(-1)^k*x^(2*k)/factorial(2*k);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$cosG6#%\"xG*()!\"\"%\"kG\"\"\")F',$*&\" \"#F,F+F,F,F,-%*factorialG6#F.F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "V tomto p\370\355pad\354 dost\341v\341me:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 116 "Int(exp(-x)*cos(x^(1/2)),x = 0 .. infinity)=Int (exp(-x)*Sum((-1)^k*x^k/factorial(2*k),k=0..infinity),x=0..infinity); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*&-%$expG6#,$%\"xG!\"\" \"\"\"-%$cosG6#*$F,#F.\"\"#F./F,;\"\"!%)infinityG-F%6$*&F(F.-%$SumG6$* ()F-%\"kGF.)F,FAF.-%*factorialG6#,$*&F4F.FAF.F.F-/FAF6F.F5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "N\341s zaj\355m\341 zda konverguje suma: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "Sum(1/factorial(2*k)*In t(exp(-x)*x^k,x=0..infinity),k=0..infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$,$-%$IntG6$*&-%$expG6#,$%\"xG!\"\"\"\"\")F/%\" kGF1/F/;\"\"!%)infinityG*$-%*factorialG6#,$*&\"\"#F1F3F1F1F0/F3F5" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 63 "Pomoc\355 methody per partes a rek urence spo\350teme integr\341l v sum\354:" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 101 "Int(exp(-x)*x^k,x=0..infinity)=[-exp(-x)*x^k]-Int( -exp(-x)*k*x^(k-1),x=0..infinity), [x=0..infinity];" }}{PARA 11 "" 1 " " {XPPMATH 20 "6$/-%$IntG6$*&-%$expG6#,$%\"xG!\"\"\"\"\")F,%\"kGF./F,; \"\"!%)infinityG,&7#,$F'F-F.-F%6$,$*(F(F.F0F.)F,,&F0F.F.F-F.F-F1F-7#F1 " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "Proto\236e je v\232ak z\341ro ve\362:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "Int(exp(-x)*x^0, x = 0 .. infinity)=1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$-%$ expG6#,$%\"xG!\"\"/F+;\"\"!%)infinityG\"\"\"" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 14 "Dost\341v\341me, \236e:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "Int(exp(-x)*x^k,x=0..infinity)=factorial(k);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*&-%$expG6#,$%\"xG!\"\"\"\" \")F,%\"kGF./F,;\"\"!%)infinityG-%*factorialG6#F0" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 33 "T\355m m\341me vyj\341d\370enu sumu ve tvaru:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 98 "Sum(Int(exp(-x)*x^k,x = 0 .. infinity)/(2*k)!,k = 0 .. infinity)=Sum(k!/(2*k)!,k = 0 .. infinity); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$SumG6$,$-%$IntG6$*&-%$expG6#,$ %\"xG!\"\"\"\"\")F0%\"kGF2/F0;\"\"!%)infinityG*$-%*factorialG6#,$*&\" \"#F2F4F2F2F1/F4F6-F%6$*&-F;6#F4F2F:F1F@" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 122 "Kter\341\236to suma na prav\351 stran\354 konverguje abs olutn\354 podle pod\355lov\351ho kriteria a nemus\355me tedy ani hleda t vhodnou majorantu." }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 7 "V\375po \350et" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 37 "Ji\236 m\371\236eme jen \+ konstatovat, \236e plat\355:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "Int(exp(-x)*cos(x^(1/2)),x = 0 .. infinity)=Sum((-1)^k*k!/(2*k)! ,k = 0 .. infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*&-% $expG6#,$%\"xG!\"\"\"\"\"-%$cosG6#*$F,#F.\"\"#F./F,;\"\"!%)infinityG-% $SumG6$*()F-%\"kGF.-%*factorialG6#F>F.-F@6#,$*&F4F.F>F.F.F-/F>F6" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 95 "podle v\354ty o z\341m\354n\354 su my a integr\341lu. A nakonec se je\232t\354 zept\341me MapleV, zda jsm e se nespletli." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 143 "if(eval f(int(exp(-x)*cos(x^(1/2)),x = 0 .. infinity),15)=evalf(sum((-1)^k*k!/ (2*k)!,k = 0 ..infinity),15)) then print(`Nespletli jste se.`) fi;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%3Nespletli~jste~se.G" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 56 "Tedy alespo\362 ne hrub\354ji ne\236li na 15 desetinn\375ch m \355st ..." }}}}}{MARK "3 5 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 } {PAGENUMBERS 0 1 2 33 1 1 }