Instructor Hentzel Office Phone: 515-294-8141 E-mail: hentzel@iastate.edu Math Department Fax: 515-294-5454 http://www.math.iastate.edu/hentzel/class.166.05 Textbook: Calculus by Varberg, Purcell, Rigdon, eight edition. March 2 10.1 Sequences p434:1,(4),11-14,(24),26,31-35,(37),(38),45 Main Idea: The calculus of power series is beautiful and gives a real understanding to some bizarre results. Key Words: Series, Sequence, Squeeze theorem, Limit, Converge Diverge, Monotone, {a } n Goal: Learn about limits of sequences. -------------------------------------------------------- Previous assignment: Page 423 Problem 6 x=Infinity x INT ----------- dx x=100 2 Sqrt[1+x ] | 2 1/2 | x=Infinity | (1+x ) | | ------------ | | (1/2)(2) | x=100 Diverges. ---------------------------------------------------- f[x_] := x/Sqrt[1+x^2]; g[x_] = Integrate[f[x],x]; a = Plot[f[x] ,{x,100,120}]; b = Plot[g[x]-g[100],{x,100,120},PlotStyle->{RGBColor[1,0,0]}]; c = Show[a,b,PlotLabel->"P423 P6 black= x/Sqrt[1+x^2]; red=INT",PlotRange->All]; Display["6.ps",c]; --------------------------------------------------------------- Page 423 Problem 10 x=1 x INT -------------- dx x=0 2 1/3 (1-x ) | 2 2/3 | x=1 (* actually, the limit as x->1 *) | (1-x ) | | --------- | | (2/3)(-2) | x=0 0 - (-3/4) = 3/4 ---------------------------------------------------- f[x_] := x/(1-x^2)^(1/3); g[x_] = Integrate[f[x],x]; a = Plot[f[x] ,{x,0,0.9}]; b = Plot[g[x]-g[0],{x,0,1},PlotStyle->{RGBColor[1,0,0]}]; c = Plot[3/4,{x,0,1},PlotStyle->{RGBColor[0,0,1]}]; d = Show[a,b,c,PlotLabel->"P423 P10 black=> x/(1-x^2)^(1/3),red=>INT", PlotRange->All,AspectRatio->Automatic]; Display["10.ps",d]; ----------------------------------------------------------------------- Page 423 Problem 16 x=3 dx INT ------------ x=0 2 x +x-2 dx 1/3 -1/3 ------------ = --------- + -------- (x-1)(x+2) x-1 x+2 f(x) = 1; g'(x) = (x+2) + (x-1) Watch out for x = 1. Why is x = -2 not a problem? | | a as a->1- | | 3 | 1/3 ln(x-1)-1/3 ln(x+2) | + | 1/3 ln(x-1)-1/3 ln(x+2) | | | x=0 | | b as b->1+ | | a as a->1- | | 3 | 1/3 ln|(x-1)/(x+2)| | + | 1/3 ln[(x-1)/(x+2)] | | | x=0 | | b as b->1+ -Infinity - 1/3 ln|-1/2| + 1/3 ln(2/5) - (-Infinity) The integral diverges. ---------------------------------------------------- f[x_] := 1/(x^2+x-2); g[x_] := (1/3) Log[Abs[(x-1)/(x+2)]]; a = Plot[f[x] ,{x,0,0.9}]; b = Plot[f[x] ,{x,1.1,3}]; c = Plot[g[x]-g[0],{x,0,0.999},PlotStyle->{RGBColor[1,0,0]}]; d = Plot[g[x]-g[0],{x,1.001,3},PlotStyle->{RGBColor[1,0,0]}]; e = Show[a,b,c,d,PlotLabel->"P423 P16 black=> 1/(x^2+x-2),red=>INT",PlotRange->All]; Display["16.ps",e]; ----------------------------------------------------------------------- Page 423 Problem 36 x=3 x INT -------- dx x=-3 9-x^2 | ln(9-x^2) |b as b->3- | --------- | | -2 |a as a->-3+ -1/2 ln(9-b^2) + 1/2 ln(9-a^2) +Infinity +(-Infinity) The integral diverges. ---------------------------------------------------- f[x_] := x/(9-x^2); g[x_] := (-1/2) Log[9-x^2]; a = Plot[f[x] ,{x,-2.9,2.9}]; b = Plot[g[x]-g[0],{x,-2.999,2.999},PlotStyle->{RGBColor[1,0,0]}]; c = Show[a,b,PlotLabel->"P423 P36 black=> x/(9-x^2),red=>INT",PlotRange->All]; Display["36.ps",c]; ----------------------------------------------------------------------- New Material A sequence is an infinite stream of numbers separated by commas. A series is an infinite stream of numbers separated by +/- signs. ____ | | Think of the tail of the |__| in se'q'uence as the comma separator. | \ ------------------------------------------------------------------ The Cake problem. Each day you eat 1/2 of the remainder of the cake. At the end of day n, how much remains? How much has been eaten. The first day you eat 1/3 of the cake. On subsequent days, you eat 1/3 of what you ate the day before. At the end of day n, how much remains? How much has been eaten? e[1] = 1/3; r[1] = 2/3; e[n_] := 1/3 r[n-1]; r[n_] := r[n-1] - e[n]; ---------------------------------------------------- e = Table[0,{i,1,20}]; e[[1]] = 1/3; r = Table[0,{i,1,20}]; r[[1]] = 2/3; Do[ (e[[n]] = 1/3 e[[n-1]]; r[[n]] = r[[n-1]]-e[[n]]),{n,2,20}]; eat = ListPlot[e]; rest = ListPlot[r,PlotStyle->{RGBColor[1,0,0]}]; y = ParametricPlot[{0,y},{y,0,1},PlotStyle->{RGBColor[0,0,1]}]; ans = Show[y,eat,rest,PlotLabel->"Cake black=ration red=remainder",PlotRange->All]; Display["eat.ps",ans]; ---------------------------------------------------- Famous sequences. 1, 2, 3, 4, 5, ..... 2, 4, 6, 8, 10, ... 1, -1, 1, -1, 1, -1 ... 1/2, 1/3, 1/4, 1/5, ... 1.1, 1.11, 1.111, 1.1111, 1.11111, ... 1, 1, 1, 1, 1, 1, 1, ... 1, 1, 2, 3, 5, 8, 13, 21, ... 1/2, 2/3, 3/4, 4/5, 5/6, ... 3, 3.1, 3.14, 3.141, 3.1415, 3.14159, ... -------------------------------------------------------------------- The Limit of a sequence. The sequences {a } is said to "converge" to L, and we write n Lim a = L n->Infinity n If for each positive number epsilon, there is a corresponding positive number N such that: n >= N implies | a - L | < epsilon n A sequence that fails to converge to any finite number L is said to "diverge". ------------------------------------------------------------------ 2 3 n Example 2 Page 431. Find Limit ---------- n->Infinity 2 7 n + 1 ------------------------------------------ f[n_] := 3 n^2 /(7n^2 + 1); A = Table[f[n],{n,1,20}]; pts = ListPlot[A]; b = Plot[3/7,{n,1,20},PlotStyle->{RGBColor[0,0,1]}]; y = ParametricPlot[{0,y},{y,0,1},PlotStyle->{RGBColor[0,0,1]}]; ans = Show[pts,b,y,PlotRange->All,PlotLabel->"Ex 2 y = 3n^2/(7n^2+1)"] Display["ex2.ps",ans]; -------------------------------------------------------------------- ln(n) Example 3 Page 431. Does -------- n->Infinity n e Use L'Hopital's rule by "extending the definition to all n. ------------------------------------------ f[n_] := Log[n]/E^n; A = Table[f[n],{n,1,20}]; pts = ListPlot[A]; y = ParametricPlot[{0,y},{y,0,1/10},PlotStyle->{RGBColor[0,0,1]}]; ans = Show[pts,y,PlotRange->All,PlotLabel->"Ex 3 y = ln(n)/e^n"] Display["ex3.ps",ans]; ------------------------------------------ The sandwich theorem: Theorem B Page 432: If {a }, {c } both converge to L, and a <= b <= c n n n n n for n >= K, (K a fixed integer), then {b } also converges to L. n ---------------------------------------------------------------- n Example 5. Show that if -1 < r < 1 then Lim r = 0. n->Infinity 1 ------ = 1 + p for some small positive number p. |r| 1 n n(n-1) 2 n(n-1)(n-2) 3 ----- = (1+p) = 1 + n p + ------- p + ------------ p + other positive stuff n 2 3 2 |r| 1 ----- > 1 + n p n |r| So 1 n ------------- > |r| 1+n p 1 Since 1+n p approaches infinity, ----- ----> 0 and so 1+n p n |r| ---> 0 by the squeeze theorem. --------------------------------------------- Sin[n] Show Limit --------- = 0. n->Infinity n ------------------------------------------ f[n_] := Sin[n]/n; A = Table[f[n],{n,1,40}]; pts = ListPlot[A]; y = ParametricPlot[{0,y},{y,0,1},PlotStyle->{RGBColor[0,0,1]}]; ans = Show[pts,y,PlotRange->All,PlotLabel->"Ex 6 y = Sin[n]/n"]; Display["ex6.ps",ans]; bns = Plot[{1/x,-1/x},{x,1,40}]; cns = Show[ans,bns, PlotLabel->"Ex 6 y=Sin[n]/n in the sandwich"]; Display["ex6x.ps",cns]; ------------------------------------------ A sequence is monotone increasing if a <= a n n+1 A sequence is monotone decreasing is a >= a n n+1 ---------------------------------------------------------------- Theorem D page 433. If U is an upper bound for a nondecreasing sequence {an} then the sequence converges to a limit A that is less than or equal to U. Similarly, if L is a lower bound for a nonincreasing sequence {bn}, then the sequence {bn} converges to a limit B that is greater than or equal to L. ------------------------------------------------------------- Examples: 3n+2 a = -------- write first four terms and find the limit. n n+1 ----------------------------------------------------------- f[n_] := (3n+2)/(n+1); A = Table[f[n],{n,1,20}]; pts = ListPlot[A]; y = ParametricPlot[{0,y},{y,0,1},PlotStyle->{RGBColor[0,0,1]}]; ans = Show[pts,y,PlotLabel->" Ex 7 an = (3n+2)/(n+1)",PlotRange->All]; Display["ex7.ps",ans]; ----------------------------------------------------------------- Example 2 a1 = 2, an+1 = 1/2 (an + ----) write first four terms and an show that it converges. ------------------------------------------------- A = Table[0,{i,1,20}]; A[[1]] = 2.0; Do[A[[n+1]] = 1/2(A[[n]] + 2/A[[n]]),{n,1,19}]; pts = ListPlot[A]; y = ParametricPlot[{0,y},{y,0,1},PlotStyle->{RGBColor[0,0,1]}]; ans = Show[pts,y,PlotLabel->" Ex 8 an+1 = 1/2(an+1/an)",PlotRange->All]; Display["ex8.ps",ans]; ------------------------------------------------- u1 = Sqrt[3] un+1 = Sqrt[3+un] Find the limit. ------------------------------------------------- A = Table[0,{i,1,20}]; A[[1]] = Sqrt[3.0]; Do[A[[n+1]] = Sqrt[3+A[[n]]],{n,1,19}]; pts = ListPlot[A]; y = ParametricPlot[{0,y},{y,0,1},PlotStyle->{RGBColor[0,0,1]}]; ans = Show[pts,y,PlotLabel->" Ex 9 un+1 = Sqrt[3+un]",PlotRange->All]; Display["ex9.ps",ans]; -------------------------------------------------