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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT 256 243 "Dr. A. Gadidov          \+
                                                         Convergence o
f sequences\nDr. R. Ablamowicz                                        \+
                                 (15 pts)                             \+
     Name\n" }}{PARA 0 "" 0 "" {TEXT -1 101 "In this lab we shall plot
 some of the terms in a sequence to better understand how sequences be
have. " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 257 10 "Example 1:" }{TEXT -1 
294 " Here is the set of commands: the first defines the sequence, the
 second evaluates a few terms in the sequence and the third one plots \+
the points. We may have a better look at the behavior of the sequence \+
for large values of  'n' by specifying the range of the values of  'a(
n)' to be dispayed." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "rest
art:a:=n->n/2^n; #define a sequence a as a function" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 60 "A:=[[n,a(n)]$n=1..100]: #specify 100 term
s of the sequence a" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "plot
(A,x=1..120, y=0..0.1,title=`Fig. 1: Sequence a(n)`,style=point); " }}
}{EXCHG {PARA 0 "" 0 "" {TEXT -1 277 "Now the picture is quite clear. \+
What conclusion can you draw? We can try to find out how far in the se
quence we must go in order to be within a given tolerance of the limit
ing value. I think that the limit of a(n) as n approaches infinity is \+
just 0. What do you think? Let L=0." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 69 "tolerance1:=0.01:L:=0:ceil(fsolve(abs(a(n)-L)=toleran
ce1,n=1..1000));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 367 "How would yo
u interpret this result? We have asked Maple to find out for what valu
e of n we have a(n) = tolerance. Maple has returned a non-integer answ
er. If you round it up to 10 (with 'ceil' function) and let N=10, then
 you will see that for all n>N we will have |a(n)| < tolerance.  Try i
t for these smaller values of the tolerance.  Wht N will be in those c
ases?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "tolerance2:=10^(-3
):L:=0:ceil(fsolve(abs(a(n)-L)=tolerance2,n=1..1000));" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "tolerance3:=10^(-4):L:=0:ceil(fsolv
e(abs(a(n)-L)=tolerance3,n=1..1000));" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 73 "tolerance4:=10^(-10):L:=0:ceil(fsolve(abs(a(n)-L)=tol
erance4,n=1..1000));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 144 "Thus, no
 matter how small tolerance is, there will be an N value such that whe
never n > N, then |a(n) - L| < tolerance.  What does it mean then?" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 141 "plot([A,tolerance1,toleran
ce2],x=1..120, y=0..0.02,title=`Fig. 2: Sequence a(n) and two toleranc
es: 0.01 and 0.001`,style=[point,line,line]);" }}}{EXCHG {PARA 0 "" 0 
"" {TEXT 258 10 "Example 2:" }{TEXT -1 30 " Let's see some more exampl
es." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "b:=n->(1-n)/(n+3);" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "B:=[[n,b(n)]$n=1..150]:L:
=-1:tolerance1:=0.2:tolerance2:=0.1:tolerance3:=0.05:" }}{PARA 0 "> " 
0 "" {MPLTEXT 1 0 122 "plot([B,L,L+tolerance1,L+tolerance2,L+tolerance
3],x=1..160,y=-1.1..0,\ntitle=`Fig. 3: Sequence b(n)`,style=[point,lin
e$4]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "ceil(fsolve(abs(b
(n)-L)=tolerance1,n=1..1000)); # see Fig. 3" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 60 "ceil(fsolve(abs(b(n)-L)=tolerance2,n=1..1000)); # \+
see Fig. 3" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "ceil(fsolve(a
bs(b(n)-L)=tolerance3,n=1..1000)); # see Fig. 3" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 113 "We can confitm our result that the limit of b(n) as
 n approaches infinity is -1 by computing this limit directly:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "Limit(b(n),n=infinity)=limit
(b(n),n=infinity);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT 259 10 "Example 3:" }{TEXT -1 102 " Let's
 try this sequence. Does this sequence have a limit? How did we explai
n this situation in class?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
23 "c:=n->(-1)^n*(1-1/n^2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
72 "C:=[[n,c(n)]$n=1..12]:plot(C,title=`Fig. 4: Sequence c(n)`,style=p
oint);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 260 10 "Example 4:" }{TEXT -1 
50 " Does this sequence have a limit? If not, why not?" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "a:=n->sin(n);" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 46 "Limit(a(n),n=infinity)=limit(a(n),n=infinity)
;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 134 "Notice that Maple says only
 that the values of sin(n) are betwee -1 and 1; it tells you that the \+
limit does not exist.  Let's see why." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 41 "NP:=100: #specify number of point to plot" }}{PARA 0 
"> " 0 "" {MPLTEXT 1 0 109 "A:=[[n,a(n)]$n=1..NP]:\nPA:=plot(A,x=0..NP
,style=point,symbol=BOX):\nPL:=plot(A,x=0..NP,style=line,color=blue):
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "plots[display](\{PA,PL
\},title=`Fig. 5: Sequence a(n) = sin(n)`);" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 261 10 "Example 5:" 
}{TEXT -1 106 " Does this sequence have a limit? If yes, why? How does
 it compare with the sequence from Example 4 above?" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 15 "a:=n->sin(n)/n;" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 46 "Limit(a(n),n=infinity)=limit(a(n),n=infinity);" 
}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "Let's see if Maple's answer is \+
correct:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "NP:=100: #speci
fy number of point to plot" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 109 "A:=[
[n,a(n)]$n=1..NP]:\nPA:=plot(A,x=0..NP,style=point,symbol=BOX):\nPL:=p
lot(A,x=0..NP,style=line,color=blue):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 65 "plots[display](\{PA,PL\},title=`Fig. 6: Sequence a(n)
 = sin(n)/n`);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 47 "We could also try our approach with tolerances:
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "NP:=80:" }}{PARA 0 "> " 
0 "" {MPLTEXT 1 0 78 "A:=[[n,a(n)]$n=1..NP]:L:=0:tolerance1:=0.06:tole
rance2:=0.04:tolerance3:=0.02:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 222 "
PL:=plot([A,L,L+tolerance1,L-tolerance1,L+tolerance2,L-tolerance2,\n  \+
            L+tolerance3,L-tolerance3],x=1..NP,y=-0.2..0.2,\ntitle=`Fi
g. 7: Sequence a(n)=sin(n)/n and tolerances: 0.06, 0.04, 0.02`,style=[
line,line$7]):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "PA:=plot(A,x=1..N
P,y=-0.2..0.2,style=point,symbol=BOX):" }}{PARA 0 "> " 0 "" {MPLTEXT 
1 0 24 "plots[display](\{PL,PA\});" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 3 "End" }}}}{MARK "40" 0 }{VIEWOPTS 1 1 0 1 1 1803 }