Mathematics by T. Arens, F. Hettlich, Ch. Karpfinger, U. Kockelkorn, K. Lichtenegger, H. Stachel (chapter 8/9: Series / Powerseries)restart; Series and PowerseriesSince the commands for evaluating partial sums and the limit of series (if they converge) have already been introduced, they are available for further examination of series and powerseries. Thus for example the alternating series yieldssum((-1)^(j+1)/j,j=1..infinity);and we may guess (slow) convergency of the partial sums from the following table (compare with Leibniz criteria/test). for n from 1 by 1 to 20 do
sn:=evalf( abs(ln(2) - sum((-1)^(j+1)/j,j=1..n) ) );
od;However, we have to be conform with convergency and divergency of series to rate the numeric results anyway. Another example are the divergent partial sums of the harmonic series. evalf(sum(1/j,j=1..100)); evalf(sum(1/j,j=1..1000)); evalf(sum(1/j,j=1..10000));The poweseries of sin and cos do have a better convergency behaviour.exp(I*Pi);for n from 1 by 1 to 12 do
evalf(sum((I*Pi)^j/j!,j=0..n));
od;We may visualize the plot of the real and imaginary part of complex valued functions.f:=(x,y)->Re(cos(x+I*y));plot3d(f,-2*Pi..2*Pi,-2..2,axes=boxed);To find an appropriate powerseries for any given function, in case it exists, the Taylorpolynominal could be determined. We are going to reconsider this command in the chapter of differential calculus. In general, Maple offer the series command, that tries to develop an expansion of any function. (If possible, the Taylorseries is determined, but even other more general expansions are applied.)series(sin(x),x=Pi/2);The expression 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 (Landau symbol) tells us, that the rest (error) of the series behaves like LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2I1EhRictRiM2Ji1JJW1zdXBHRiQ2JS1GLDYlUSJ5RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW5HRiQ2JFEiNkYnL0Y7USdub3JtYWxGJy8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRicvJSdmYW1pbHlHUTBUaW1lc35OZXd+Um9tYW5GJy8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRidGQUYrRkZGSUZB , if 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 , so that by 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 with a constant 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 the error could be estimated.series(x^x,x=0); Exercises(To work on the exercises, please open a new worksheet to trial an error your commands. You will get a suggestion for the solution, if you open up the solution. Before you look at them try on your own.)1. Determine the limit of the series LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2I1EhRictRiM2KS1JK211bmRlcm92ZXJHRiQ2Jy1JI21vR0YkNi1RJiZTdW07RicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y9LyUpc3RyZXRjaHlHUSV0cnVlRicvJSpzeW1tZXRyaWNHRj0vJShsYXJnZW9wR0ZCLyUubW92YWJsZWxpbWl0c0dGQi8lJ2FjY2VudEdGPS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjE2NjY2NjdlbUYnLUYjNigtRiw2JVEibkYnLyUnaXRhbGljR0ZCL0Y5USdpdGFsaWNGJy1GNTYtUSI9RidGOEY7Rj4vRkFGPUZDL0ZGRj0vRkhGPUZJL0ZMUSwwLjI3Nzc3NzhlbUYnL0ZPRltvLUYsNiVRIm1GJ0ZWRlgvJSdmYW1pbHlHUTBUaW1lc35OZXd+Um9tYW5GJy8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRidGOC1GLDYlUSgmaW5maW47RidGVkZYLyUnYWNjZW50R0Y9LyUsYWNjZW50dW5kZXJHRj1GKy1JJ21zcGFjZUdGJDYmLyUnaGVpZ2h0R1EmMC4wZXhGJy8lJndpZHRoR1EkNS4wRicvJSZkZXB0aEdGYnAvJSpsaW5lYnJlYWtHUSVhdXRvRictRiw2JVEkYV9uRidGVkZYRmBvRmNvRjhGK0Zgb0Zjb0Y4 with a) m=1, 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) m=0, 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c) m=1, 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d) m=1, 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e) m=0, 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f) m=3, 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 . SolutionWe obtainsum((-1)^j/(j*(j+4)),j=1..infinity);
evalf(%);with the partial sumsfor n from 1 by 1 to 6 do
evalf(sum((-1)^j/(j*(j+4)),j=1..n),5);
od; and sum(1/(j!)^2,j=0..infinity);
evalf(%);
for n from 0 by 1 to 5 do
evalf(sum(1/(j!)^2,j=0..n),5);
od;Note, the predefined special function Bessel (here e.g. a Bessel function). Dealing with series, such functions are going to reappear frequently. In exercise c)-e) the result issum(1/(j*(j+1)),j=1..infinity);
sum(1/j^j,j=1..infinity);
sum(j/(2*j+1),j=0..infinity);
Likewise we expect, Maple cannot explicitly evaluate a limit in every situation. For the last example we obtainsum((2*j+6)/(j*(j+1)*(j+2)),j=3..infinity);2. Determine the radius of convergency for the 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 und 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 . SolutionWe are going to try the ration test. First we define a sequence of the absolute values of the summands for example as a function of n a:=n->1/sqrt(n+1)*abs(x)^n;and we obtain (after resetting n)n:='n':
limit(a(n+1)/a(n),n=infinity);Therefore by the ratio test r=1 is the radius of convergency around the center of expansion x_0=0.In the second example we may once again apply the ratio test or we directly compute the radius of convergency r by the root test in the surrounding of x_0=1r:= limit(1/((n+1)/2^n)^(1/n),n=infinity);3. Plot the imaginary part of the complex logarithm function. Solutiong:=(x,y)->Im(ln(x+I*y));plot3d(g,-2..2,-2..2,axes=boxed);If you rotate the plot, you can observe the discontinouity at the negative real axis.