Sequences and SeriesLast Revision: October 3, 2010
<Text-field style="Heading 1" layout="Heading 1">Sequences</Text-field>Throughout we shall use the example 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 we want to calculate specific members of this sequence. To do so, it is easiest to define the sequence as a functiona := n -> (3*n^2 + (-1)^n*n*cos(2*n))/(n^2+1);Computation of say, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMUYnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y+ or LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEjMTJGJy9GNlEnbm9ybWFsRidGMkY1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGPg== , is now very simple.a(1); a(12);and in decimals we haveevalf(a(1)); evalf(a(12));Limits of sequences are done the same way they are done for functions, either directly as in limit( a(n),n=infinity);or in the two-step method (recommended)Limit( a(n),n=infinity);
value(%);Of course, the two results are identical. You can also use the template from the Expressions button on the left.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Graphical RepresentationSequences are functions which are defined for natural number inputs on, and in our example the 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 term prevents us from plotting the sequence like we would a function. A better way to depict sequences is to plot them as points of the form 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's say we want to plot the first twenty points of the sequence. In the first step we write these 20 points as ordered pairs into an array using the seq command (output suppressed).P := seq( [n, a(n)], n=1..20):For instance, the 12th point in this array isP[12];
evalf(%);Now we display all twenty points in a graph.plot( [P], style=point, view = [0..20,0..4]);The limit of 3 is apparent.
<Text-field style="Heading 1" layout="Heading 1">Recursive Sequences</Text-field>Here we use the example 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with 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 initial value.First we use a loop to compute the first ten terms of this sequence:a[1] := 0;
for n from 1 to 9 do
a[n+1] := 2 + 1/(2+a[n]);
end;Now we use the seq command to generate the points for the plotpts := seq( [n,a[n]], n=1..10);We are now ready to plot the sequence.plot( [pts],style=point,view=[0..10,0..4]);The graph shows that the sequence converges rather quickly. In order to find the limit, we replace 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and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRIm5GJ0YyRjVGMkY1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRictSSNtb0dGJDYtUSJ+RicvRjZRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZILyUpc3RyZXRjaHlHRkgvJSpzeW1tZXRyaWNHRkgvJShsYXJnZW9wR0ZILyUubW92YWJsZWxpbWl0c0dGSC8lJ2FjY2VudEdGSC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRldGRA==by LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiTEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIn5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTEY5in the recurrence relation, and then solve the resulting equation for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiTEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=. L = 2+1/(2+L);
solve(%,L);Our limit is the positive square root. The computationsevalf( a[8]); evalf(sqrt(5));show that the term LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiOEYnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy1JI21vR0YkNi1RIn5GJ0Y+LyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZJLyUpc3RyZXRjaHlHRkkvJSpzeW1tZXRyaWNHRkkvJShsYXJnZW9wR0ZJLyUubW92YWJsZWxpbWl0c0dGSS8lJ2FjY2VudEdGSS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlhGPg==is already very close to the limit.
<Text-field style="Heading 1" layout="Heading 1">Finite Sums</Text-field>restart;We begin by looking at finite sums. These can be computed with the sum command, or by using the template on the left.Example 1: We calculate the first 12 terms in the harmonic series.sum( 1/n, n=1..12);It is recommended that you use a two step procedure:Sum( 1/n, n=1..12);
value(%);The numerical value is found with evalfevalf(%);Using the template instead this calculation 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 2: Add the first n natural numbersSum( k , k=1..n);
value(%);
factor(%);Example 3: Add the squares of the first n natural numbersSum( k^2 , k=1..n);
value(%);
factor(%);Example 4: Add 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.Sum( 3^n,n=0..10);
value(%);
<Text-field style="Heading 1" layout="Heading 1">Series (Infinite Sums)</Text-field>For infinite sums (series) we use infinity as the upper limit of summation.Example 1: Compute 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Sum(1/n^2, n = 1 .. infinity);
value(%);Example 2: Evaluate the alternating harmonic series.Sum( (-1)^(n+1)/n,n=1..infinity);
value(%);
evalf(%);Example 3: The series LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkrbXVuZGVyb3ZlckdGJDYnLUkjbW9HRiQ2LVEmJlN1bTtGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdRJnVuc2V0RicvJSpzZXBhcmF0b3JHRjcvJSlzdHJldGNoeUdRJXRydWVGJy8lKnN5bW1ldHJpY0dGNy8lKGxhcmdlb3BHRjwvJS5tb3ZhYmxlbGltaXRzR0Y8LyUnYWNjZW50R0Y3LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMTY2NjY2N2VtRictRiM2Ji1JI21pR0YkNiVRIm5GJy8lJ2l0YWxpY0dGPC9GM1EnaXRhbGljRictRi82LVEiPUYnRjIvRjZRJmZhbHNlRicvRjlGWS9GO0ZZL0Y+RlkvRkBGWS9GQkZZL0ZERlkvRkZRLDAuMjc3Nzc3OGVtRicvRklGW28tSSNtbkdGJDYkUSIxRidGMkYyLUYjNictRi82LVEoJmluZmluO0YnRjJGWEZaRmVuRmZuRmduRmhuRmluRkUvRklGR0ZRLyUrZm9yZWdyb3VuZEdRLFsyMDAsMCwyMDBdRicvJSxwbGFjZWhvbGRlckdGPEZTRmluLyUsYWNjZW50dW5kZXJHRlktSSZtZnJhY0dGJDYoRk0tRiM2JS1JJW1zdXBHRiQ2JS1GXm82JFEiMkYnRjItRiM2JUZNRlFGUy8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRidGUUZTLyUubGluZXRoaWNrbmVzc0dGYG8vJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGYnEvJSliZXZlbGxlZEdGWS1GLzYtUSJ+RidGMkZYRlpGZW5GZm5GZ25GaG5GaW5GRUZmb0Yy converges by the ratio test. The result is surprisingly simple.Sum( n/2^n , n=1..infinity);
value(%);Example 4: This series 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 must converge by the Alternating Series Test. The result is a mess, a decimal approximation is more useful.Sum( (-1)^(n+1)*n^2/(n^3+1), n=1..infinity);
value(%);evalf(%);It appears that maple had to resort to complex numbers to find a closed form expression for the value of the series. The conversion to decimals left a small contribution from imaginary numbers, which cannot be part of the correct answer, since the original sum does not contain complex numbers.
Example 4: The geometric series formula is built in.Sum( r^n, n=0..infinity);
value(%);Example 5: The series 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 converges absolutely, but maple can only compute its value using special functions. Decimal approximation shows that the sum is slightly larger than 1.Sum( sin(n)/n^2,n=1..infinity);
value(%);
evalf(%);Here is a plot of the first 30 partial sums for this sum.P := seq([n, sum(sin(k)/k^2,k=1..n)],n=1..30):
plot([P], style=point, view=[0..30,0.9..1.1]);LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=