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.ICN Textbook: Calculus Early Vectors Preliminary Edition By Stewart. Monday, April 28 These last three days of classes will be over series. There are no hand-in-homework and no tests. This material is part of the AP Calculus curriculum. The syllabus is: Mon, Apr 28 10.1, 10.2, Sequence, Series, Harmonic Series, 10.3, 10.4 Geometric Series, Ratio Test, Root Test Comparison Test, Wed, Apr 30 10.5, 10.6 Power series, Radius of Convergence, Term by term Differentiation and Integration Fri, May 2 10.7 MacLaurin Series, Taylor Series, Error Bounds --------------------------------------------------------------- Main Idea: You can make a cake last forever by taking small bites. Key Words: Sequence, Series, Harmonic series, Geometric Series, Ratio Test, Root test. Goal: Learn series techniques as a means to solve problems. ------------------------------------------------------------ The difference between a series and a sequence of numbers is that the series is separated by plus signs, and the sequence is separated by commas. (* think of the "tail" of the q as reminding you that it is sequence that has the commas *) 1, 2, 3, 4, ... n, ... is a sequence. 1, 1/2, 1/3, 1/4, ..., 1/n, ... is a sequence. 1, -1, 1, -1, 1, -1, ..., (-1)^n is a sequence. 1+2+3+4+ ... + n + ... is a series 1+1/2+1/3+1/4+ ... + 1/n + ... is a series. 1-1+1-1+1-1...+(-1)^n+ ..... is a series. --------------------------------------------------------- You get a birthday cake. You want to make it last a long time so each day you eat just 1/2 of what remains. How long will the cake last? ____________________________________________ | | | | | | | | | | | | | 1/32|____| | | | | | | | 1/8 | |1/64| | | |_____|____| | | | | | | | | | 1/2 | | 1/16 | | | | | | |__________|__________| | | | | | | | | | | | | | | 1/4 | | | | | | | | | | |____________________|_____________________| The amount eaten will be the sum 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64 + ... + 1/2^n + .... Certainly, you can see that the result will be that you eventually will eat the entire cake. The upper right hand corner will not be eaten, but that is on the pan, anyway. -------------------------------------------------------- What if you choose only to eat 1/3 of the remaining cake every day. How much of the cake will you eat. eat leave 1/3 2/3 2/9 4/9 4/27 8/27 1/3 + 2/9 + 4/27 + ....+ (1/3)(2/3)^n + ........ Since the residue shrinks to zero, you will also eat the whole cake. -------------------------------------------------------- What if you base the amount you eat on the serving size. Each day you eat 1/3 of the amount you ate the day before? 1/3 + 1/9 + 1/27 + ... + (1/3)^n + ..... Now you do not eat the whole cake. You only eat half of it. You can visualize your roommate eating from the other half of the cake. Each time you divide the remaining cake into thirds and jointly consume 2/3 of it. Your share in total will amount to 1/2. ________________________________________________________________ | | : : | | | | : : | | | | : : | | | | : : | | | | : : | | | | : : | | | 1/3 | 1/9 : : (1/9)| (1/3) | | | : : | | | | : : | | | | : : | | | | : : | | | | : : | | | | : : | | | | : : | | |____________________|____________________|____________________| ---------------------------------------------------------------------- The formula for a geometric series is a a + ar + ar^2 + ar^3 + ... + ar^n + ... = ---------- 1-r The proof requires n x - 1 n-1 ----------- = 1 + x + x^2 + ... + x x-1 n 2 n-1 a(r - 1) Thus a + ar + ar + ... + ar = -------- r-1. n a If |r| < 1, then r vanishes and the limit will be -------- 1-r ----------------------------------------------------------------- Find 3 + 3/5 + 3/25 + 3/125 + ... + 3/5^n + ... 3 15 --------- = -------- = 15/2 1-3/5 5-3 ---------------------------------------------------- The Harmonic Series is 1 + 1/2 + 1/3 + 1/4 + ... 1/n + ... It does not converge. If you use enough terms it will be infinitely large. It is the solution of the domino problem. How far can a stack of dominos over hang. The answer is infinitely far. __________________ [_________|________] __________________ [_________|________] 1 __________________ [_________|________] 1/2 __________________ [_________|________] 1/3 __________________ [_________|________] 1/4 __________________ [_________|________] 1/5 __________________ [_________|________] 1/6 . . With this over hang, the center of gravity of the stack of dominos above the current level will be just over the end of the next domino below. -------------------------------------------------------- It is also the solution to the wave propagation problem. If you send a square wave pulse, the tails die off as 1/n. This means that you can always get an echo of any size. The better way to shape the pulse is by a raised cosine curve. Then the tails are like (1/n)^2 which is convergent. If you send slow enough, you will never have false echo problems. ---------------------------------------------------------- Proof that the harmonic sum does not converge. S = 1+1/2+1/3+1/4+1/5+1/6+1/7+1/8+1/9+...1/16+1/17+...+1/32+ - --- ------- --------------- ----------- ------------ 1 1/2 >1/2 > 1/2 > 1/2 > 1/2 S >= 1 + 1/2 + ..... + 1/2^n = 1 +n/2 for any n. ---------------------------------------------------------- Puzzle ----------------------------------------------- S = 1-1/2+1/3-1/4+1/5-1/6+1/7-1/8+1/9+...-(-1)^n/n + ... 2S = 2-2/2+2/3-2/4+2/5-2/6+2/7-2/8+2/9+...-2(-1)^n/n + ... _ | -2/2 -2/4 -2/6 -2/8 -2/10 -2/12 -2/14 - ... |_ 2 +2/3 +2/5 +2/7 +2/9+ 2/11 +1/13 + 2/15 + .... _ | -1 -1/2 -1/3 -1/4 -1/5 -1/6 -1/7 - ... |_ 2 +2/3 +2/5 +2/7 +2/9+ 2/11 +1/13 + 2/15 + .... 1 -1/2 + 1/3 -1/4 + 1/5 -1/6 + 1/7 .... Thus 2 S = S so must S = 0? No, It is not proper to rearrange the terms in a series. Actually, S = ln(2). ------------------------------------------------------------------ If ao + a1 + a2 + ... + an + ... is a series and |ao|+|a1|+|a2|+|a3|+...+|an|+ ... converges, we say the sequence is absolutely convergent. If the sequence is absolutely convergent, then it is convergent. If the sequence is convergent but NOT absolutely convergent, we say the sequence is conditionally convergent. The Harmonic series is the standard example of a conditionally convergent series. ------------------------------------------------------------ The proof the absolutely convergent implies convergent. |Infinity | Infinity Proof: | SUM ai | <= SUM |ai|. | i=n | i=n If the series is absolutely convergent then the right hand sum can me made a small as one desires by making n huge. Thus the left hand sum can also be made as small as one desires. ------------------------------------------------------------ There are tests to show when a series converges. Certainly, it is necessary for the nth term to go to zero. But that is not enough. The harmonic series is the standard example of a series whose nth term goes to zero which does not converge. -------------------------------------------------------------- The ratio test: L < 1 then the series converges. If Limit |an+1/an| = L L > 1 then the series diverges. n->infinity L = 1 Cannot say one way or the other. --------------------------------------------------------------------- Show that 2^n SUM ------- converges. n! a n+1 2^(n+1) n! 2 -------- = --------- -------- = -------- a (n+1)! 2^n n+1 n a n+1 Limit ------ = 0. Therefore the series converges. n->Infinity a n ------------------------------------------------------------ Sqrt[n] Show that SUM ----------- Converges. 1+n^2 a n+1 Sqrt[n+1] 1+n^2 ---------- = ---------- -------- a 1+(n+1)^2 Sqrt[n] n a n+1 n+1 1+n^2 Limit ------- = Limit Sqrt[-----] ----------- = 1 a n 1+(n+1)^2 n You cannot tell if this series converges or diverges using the ratio test. -------------------------------------------------------- Infinity 1+2^n SUM -------- n=1 1+3^n a n+1 1+2^(n+1) 1+3^n -------- = ---------- -------- a 1+3^(n+1) 1+2^n n a n+1 1+2^(n+1) 1+3^n Limit -------- = Limit ---------- ----------- = 2/3. n->Infinity a 1+2^n 1+3^(n+1) n The series converges. ---------------------------------------------------- The root test. _____________ n/ if L < 1 the series converges Let Limit \/ | an | = L. if L > 1 the series diverges if L = 1, it can go either way. --------------------------------------------------------------- The proof of the ratio test: Suppose that Limit |an+1/an| = r < 1. n->Infinity Pick any s such that r < s < 1. From some no on, |an+1/an| < s. (* i.e. |an+1| < s |an| *) Term by term the series is bounded by the harmonic series |ano| + |ano| s + |ano| s^2 + .... + |ano| s^n + ... and is absolutely convergent. Thus the original series is convergent. ------------------------------------------------------------------ Assignment for fun. Zeno's Paradox. Achilles chases a tortoise. The tortoise has a 1 mile head start. When Achilles reaches the spot the tortoise was, the tortoise has moved 1/10 mile further. When Achilles reaches the new spot, the tortoise will have moved 1/100 mile further. When and where will Achilles catch the tortoise. ---------------------------------------------------------------- The hand of the clock are together at 12:00. When the hour hand gets to 1, the minute hand is on the 12. When the minute hand moves to the 1, the hour hand has moved further away. When the minute hand catches up the hour hand has moved ever so tiny much further. When will the minute hand catch the hour hand? --------------------------------------------------------------- Sue needs a medicine in her body. During each day her metabolism reduces the amount of medicine in her body by 60 %. That is, only 40% of what is in her body today will be there tomorrow. Each pill has 10 mg of medicine. There must be 16 mg in her body for the medicine to be effective. How many doses till this level is reached? The medicine has moderate side effects when the dose in the body reaches 16.6 mg. How many doses till this is reached? The medicine has horrible side effects when the medicine in the body reaches 16.7 mg. How many doses till this is reached? ------------------------------------------------------------------ For the snowflake curve. (a) What is the length of the perimeter. (b) What is the area. ------------------------------------------------------------------- Try proving the root test. ------------------------------------------------------------------- m = Table[0,{i,1,100}]; m[[1]] = 1; Do[m[[n]] = 0.4 m[[n-1]] + 1,{n,2,100}]; n = Table[ 0.4 m[[n]],{n,1,100}]; a = ListPlot[m,PlotStyle->{RGBColor[1,0,0]}]; b = ListPlot[n,PlotStyle->{RGBColor[0,0,1]}]; c = Show[a,b,PlotLabel->"Medicine"]; Display["pill.ps",c];