(*
CALCULUS 265 Honors
Fall Semester 2000
8:00 to 8:50 MTTF
Carver 0002

Instructor: Hentzel
Office: 432 Carver
Office Hours: 9:00-10:00 MTTF

E-mail  hentzel@iastate.edu
phone  294-8141
Website

http://www.math.iastate.edu/hentzel/honors.265

Thursday, November 16:    Section  8.5

Previous assignment
p587:38,39,45,48,54,57,59,60

Main idea: Exploring the beauty of sequences. 

Key words: Alternating Series, Absolute Convergence,

Power series, Differentiating term by term, Circle of

Convergence, Conditional Convergence, Rearrangement of Series, 

Remainder

Goals: Raise your conscience level to possibilities of series. 

Previous Assignment: 

Compare the Cosine curve to the approximating polynomials

of degree 4, 6, 8, 10 over the integral [0,2 Pi].  Plot out

the graphs.  What degree would be necessary to assure an

accuracy of less than 0.0001 on the interval [0,2 Pi]?
*)

f4[x_]  := Sum[ (-1)^n x^(2 n)/(2 n)!,{n,0,2}];
f6[x_]  := Sum[ (-1)^n x^(2 n)/(2 n)!,{n,0,3}];
f8[x_]  := Sum[ (-1)^n x^(2 n)/(2 n)!,{n,0,4}];
f10[x_] := Sum[ (-1)^n x^(2 n)/(2 n)!,{n,0,5}];
f24[x_] := Sum[ (-1)^n x^(2 n)/(2 n)!,{n,0,12}];

 a4 = Plot[ f4[x],{x,0,2 Pi}];
 a6 = Plot[ f6[x],{x,0,2 Pi}];
 a8 = Plot[ f8[x],{x,0,2 Pi}];
a10 = Plot[f10[x],{x,0,2 Pi}];

a12 = Plot[Cos[x],{x,0,2 Pi}];

ans = Show[a4,a6,a8,a10,a12];

ans1 = Show[ans,PlotLabel->"Degree 4,6,8,10 Approximations to Cosine"]

ans2 = Plot[{f24[x],Cos[x]},{x,0,4 Pi}];

(*
How many terms are necessary to approximate Sin[n] to within 0.0001 on [0,2 Pi]?





f[n_] := N[ (2 Pi)^n /n!]      The degree n term of alternating decreasing 
series for Sine and Cosine.

1 6.28319           2 19.7392
3 41.3417           4 64.9394
5 81.6052           6 85.4568
7 76.7059           8 60.2446
9 42.0587          10 26.4263
11 15.0946         12 7.90354
13 3.81995         14 1.71439
15 0.718122        16 0.282006
17 0.104229        18 0.0363828
19 0.0120316       20 0.00377983
21 0.00113092      22 0.000322991
23 0.0000882353    24 0.0000231  <++++++++++++++  Use polynomial of degree 22
                                   for [0,2 Pi].

If we only want it to be accurate on [0,Pi/2] then we can use fewer terms.
In[12]:= Do[Print[n," ",f[n]],{n,1,10}]
1 1.5708
2 1.2337
3 0.645964
4 0.25367
5 0.0796926
6 0.0208635
7 0.00468175
8 0.00091926
9 0.000160441
10 0.000025202  <+++++++ Use polynomial of degree  8 for [0,Pi/2]

*)



e^x  = 1 + x + x^2/2 + x^3/3! + x^4/4! + ... + x^n/n! + ...

Sin[x] = x - x^3/3! + x^5/5! - x^7/7! + ...

Cos[x] = 1 - x^2/2! + x^4/4! - x^6/6! + ...

In Class experience.
-------------------

(a)  Show that e^(ix) = Cos[x] + i Sin[x].

(b)  Find Log[i].


Find a power series for

        1
(a)  --------
       1-x

      
         1
(b)  ----------
       1+x^2


          x
(c)   ---------
        1+x^2

          1
(d)     ------ 
        (1-x)^2

What is the radius of convergence for all of the above series?