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.166.08 Textbook: Calculus by Varberg, Purcell, Rigdon, ninth edition. Monday, March 31 9.6 Power series p483: 2, 6,12,24 Main Idea: Power series are polynomials of infinite degree. They behave like polynomials as long as you remember to avoid like the plague the places that they do not converge. Key Words: (a) Radius of Convergence, (b) Interval of Convergence, (c) Power Series, Goal: For series of the form: 2 3 n f(x) = a + a x + a x + a x + ... + a x + ... 0 1 2 3 n It is quite easy to visualize where they converge. -------------------------------------------------------- Previous assignment: p505: 1-28 Give Reasons See March.28.answers --------------------------------------------------------------------------- I want to take a few moments on the question: "What is a number". A related question is: what does it mean to say: "I know the way to drive to Iowa City." To one person, he knows that Iowa City is 100 miles east and 30 miles south of Ames, so he knows how to get there. To another person, knowing the way means that he must know every highway name, every corner, and every intermediate town. To the first person, there is not one "way" to Iowa City. One can go many various ways. But he will have no knowledge of places to stop, road conditions, or any local points of interest. The second person will be able to plan his route with all the comforts possible. But if there is a detour and he has to leave his standard route, or has to get to Iowa City from a different starting point, he may first have to drive to Ames. Which is the "better" navigational system? =============================================================== New Material: In Euclidean geometry, you think that the angle of 20 degrees exists. You can approximate it as closely as you want. But because you can not construct it exactly, you are not allowed to say that it exists. In calculus if you can tell how to approximate something, that means you know that number and can use it in any calculations you want. 2 Infinity 1 Pi We know that SUM ---- = ----- n=0 2 6 n Infinity 1 We know that SUM ----- converges, but we do n=0 3 n not know what it converges to. This means that we do not know how to express the limit. Of course, we can just refer to its limit as "that number that the p-series, p=3 converges to". But if we have another number expressed like that, can we really say whether they are equal, or which one is larger? This is supposed to be a property of real numbers. It is called trichotomy. Given two numbers a and b, exactly one of these things hold. ab. ------------------------------------------------------------------ This concept is really used in power series. Even if we do not know what a series converges to, we can still take its derivatives and its integrals. These will again be power series. We can write down a power series which is the solution to a problem and then we can claim that we have the answer because the answer will be what the series converges to. ------------------------------------------------------------------ For a series 2 3 n f(x) = a + a x + a x + a x + ... + a x + .... 0 1 2 3 n (a) Where does it converge? -------------------|##############################|------------------- diverges /|\ Converges absolutely /|\ diverges | | | | | | | | At the interface the series may converge, may diverge, or may converge conditionally. / (b) f (x) is obtained by differentiating the power series term by term. (c) INT f(x) is obtained by integrating the power series term by term. Conclusion, If you allow yourself the luxury of working with power series, even the most complicated problem is reduced to polynomials. ------------------------------------------------------------------ Page 480 Example 1. 2 3 n f(x) = a + a x + a x + a x + .... + a x ..... What is its limit. Where does it converge. f(x) is a geometric series, p = x and it converges if |x| < 1. a f(x) = ------ 1-x a f(x) = -------- 1-x ----------------|#################################|------------------ -1 1 diverges diverges ----------------------------------------------------------------- Get["font.math"]; a = 3; s[n_,x_] := Sum[a x^i,{i,1,n}]; p1 = Table[ Plot[ s[h,x],{x,-0.99 ,0.9 }],{h,5,30,5}];; p2 = Plot[ a x/(1-x),{x,-1.1,0.99},PlotStyle->{RGBColor[1,0,0]}]; p3 = Show[p1,p2,PlotLabel->"Page 480 Ex 1: SUM a x^i"]; Display["ex1.ps",p3]; ----------------------------------------------------------------- Page 480 Example 2. n 2 3 Infinity x x x x f(x) = SUM ----------- = 1 + --- + ---- + ---- + ... n=0 n 2 3 (n+1) 2 2 2 3 2 4 2 n x a = ----------- n n (n+1) 2 a n+1 n n+1 x (n+1) 2 (n+1) x (a) ------- = -------------- * ------------ = --------------- =====> x/2 a n+1 n (n+2) 2 n (n+2) 2 x (b) The series converges by the ratio test when |x| < 2. diverges converges absolutely diverges ----------------|#################################|------------------ - 2 2 conditional diverges convergence At x = -2 the series is the alternating harmonic series and converges conditionally. At x = 2 the series is the harmonic series and diverges. ------------------------------------------------------------------- Get["font.math"]; f[x_,n_] := x^n/((n+1) 2^n); g[n_] := Plot[Sum[f[x,i],{i,1,n}],{x,-2.3,2.1}]; H = Table[g[2 n],{n,1,20}]; a = Plot[Sum[f[x,i],{i,1,200}],{x,-2,2}, PlotStyle->{RGBColor[1,0,0],Thickness[0.01]}]; b = Plot[0,{x,-2,2},PlotStyle->{RGBColor[0,0,1],Thickness[0.01]}]; c = Show[H,a,b,PlotLabel->"P480 Ex 2; an = 2 x^n/((n+1) 2^n) ", PlotRange->All]; Display["ex2.ps",c]; ------------------------------------------------------------------- Page 480 Example 3 n Infinity x f(x) = SUM -------- n=0 n! n x a = -------- n n! | a | n+1 | n+1 | |x| n! |x| (a) | ----- |= -------- * ------ = ------------ ======> 0 | a | (n+1)! n n+1 | n | |x| (b) by the ratio test, r = 0, the series converges for all x. Converges absolutely <###########################################################################> ============================================================================== Get["font.math"]; f[x_,n_] := x^n/n! ; g[n_] := Plot[Sum[f[x,i],{i,1,n}],{x,-3, 3}]; H = Table[g[2 n],{n,1,20}]; a = Plot[Sum[f[x,i],{i,1,200}],{x,-3,3}, PlotStyle->{RGBColor[1,1,0],Thickness[0.01]}]; b = Show[a,H,PlotLabel->"P480 Ex 3; an = SUM x^n/n! ",PlotRange->All]; Display["ex3.ps",b]; =========================================================================== ------------------------------------------------------------------------- The proof of convergence is very very very clever. We simply show that if the series converges at x = b, then it converges absolutely for |x| < |b|. Infinity n Proof: Suppose that SUM a b converges. n=1 n n Then Limit a b = 0 so there exists an N such that n > N n->Infinity n 0 0 | n | implies |a b | < 1. | n | _ _ n _ _ n Infinity | || |n Infinity | n | | |x| | Infinity | |x| | (a) Now SUM |a ||x| = SUM |a b | | --- | < SUM | ----| n=1 | n || | n=1 | n | |_|b|_| n=1 |_|b|_| for n > No (b) The last series is a geometric series with p = |x|/|b| which converges if |x| < |b|. (c) Therefore the original series converges by the comparison test. ================================================================================ Write the power series for 1 (a) f(x) = ------ 1-x 1 (b) f(x) = ----- 1+x x (c) f(x) = ---------- 2 1+x (d) f(x) = ln(1+x) x (e) f(x) = e