{VERSION 4 0 "IBM INTEL LINUX" "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 }{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 }} {SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "#sfb 27 mar 01" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "# Lets show the taylor serie s for cos(x) converges to something." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "taylor(cos(x),x,10);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "sum((-1)^i*x^(2*i)/(2*i)!,i=0..5);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "Sum((-1)^i*x^(2*i)/(2*i)!,i=0..5);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "approx:=Sum((-1)^i*x^(2*i)/( 2*i)!,i=0..N);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "# the que stion is to find an estimate for the error:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "Sum((-1)^i*x^(2*i)/(2*i)!,i=0..infinity)-approx;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "#This doesn't look like m uch help, but lets cancel the earlier terms leaving us the tail" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "Sum((-1)^i*x^(2*i)/(2*i)!,i= N+1..infinity);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 126 "# lets \+ think of x as |x| and so x is positive, this error would be bigger if \+ all the terms were positive. Our estimate becomes" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 36 "Sum(x^(2*i)/(2*i)!,i=N+1..infinity);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 121 "#We want to find a geometri c series which dominates the above series and converges (to something \+ we know how to compute)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "a:=x^(2*(N+1))/(2*(N+1))!;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "r:=(x^(2*(N+2))/(2*(N+2))!)/a;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(r);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "#note r < 1 if x < 2*N and r < 1/4 if x < N." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 127 "#note also that the ratios of the terms ar e decreasing as N increases as long as x < 2*N. Thus we have (using A \+ for a, R for r)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "Sum(x^(2 *i)/(2*i)!,i=N+1..infinity) " 0 "" {MPLTEXT 1 0 58 "#And (we can assume that x < N so r < 1/4 so 1/(1-r) < 4/3" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 " Sum(A*R^i,i=0..infinity)=A/(1-R);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "Sum(x^(2*i)/(2*i)!,i=N+1..infinity)< 4*a/3;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "#The final observation is th at " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "Limit(x^n/n!,n=infin ity)=limit(x^n/n!,n=infinity);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 261 "# so the infinite series does converge. (But we did not show \+ that it converges to cos(x). One needs to look at the remainder term o f taylor's polynominal to do this. The remainder term is an alternate \+ way of showing the series as well. But that is in the text.)" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with(DEtools);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "a:=dfieldplot(diff(y(t),t)=y(t)*(4- y(t)),y(t),t=0..5,y=-1..5,arrows=LINE):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "sol:=4*A*exp(4*t)/(1+A*exp(4*t));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 113 "b:=plot([subs(A=4/7,sol),subs(A=1/7,sol),0 ,4,subs(A=-1,sol),subs(A=-5,sol),subs(A=-1/5,sol)],t=0..5,color=black) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "display(a,b);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "0 0 0" 14 }{VIEWOPTS 1 1 0 3 2 1804 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }