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Exercise 3. (a) Use the procedures S and Spts to investigate the limit of the partial sums 1 1 Sn = 1 + + . . + 2 4 n as n → ∞. Does the limit exist? That is, do the partial sums approach some well-defined number or do they grow without bound? (b) Repeat for the harmonic series 1+ 1 1 + ... + + ... 2 n (c) Plot the partial sums for both series together: > plot( {Spts(x ->1/x, 100), Spts(x->1/x^2, 100)}); Exercise 4. Study the alternating harmonic series 1− 1 1 1 1 + − + ... ± + ... 2 3 4 n Does it converge?

8 Programming in Maple Our loop as written prints out all the intermediate sums. 0/i^2: od: total; Warning: Only one Maple prompt (>) appears in the loop. This is the safest way to type in a loop because it ensures that every time you start a loop, you also end it. A command like > for i from 1 to 5 do > i^2; > od; works, but can easily lead to disaster. For example, you might start typing a loop and realize that you forgot to do something else first, say define a function that you need inside the loop.

0/x^2; Then S( f, 4 ) gives the sum. 0/x^2, 4 ); To sum up a different series, we just change the arguments to S. For example, > S( i -> 1/i, 10 ); sums the first ten terms of the harmonic series: 1+ 1 1 1 + + ... + + ... 2 3 n Lists Another way to think about the behavior of an infinite sum is to plot the partial sums Sn versus n. This gives a picture of how the sum grows as n increases. We can modify our procedure S to produce a list of points (n, Sn ) and then use plot to graph them. To define our list we use Maple’s sequence-building command.