New PDF release: Multivariable Mathematics with Maple: Linear Algebra, Vector

By James A. Carlson (Author), Jennifer M. Johnson (Author)

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Extra info for Multivariable Mathematics with Maple: Linear Algebra, Vector Calculus and Differential Equations

Sample text

It consists of the key words do and od (that is, do spelled backwards) enclosing a list of instructions. In our case we have only one instruction — add the quantity 1/i^2 to the current value of total and then store the result in total. Running our loop is the same as executing the sequence of commands below. 0/i^2; Consequently, the counter i has the value 5 when the loop is complete. This may cause trouble later, so you might want to clear the variable i with i := ’i’ when you have finished the problem.

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? If so, does it converge quickly or slowly? Compare with the series 1 1 1 1 + ... ± 2 + ... 1− + − 4 9 16 n 33 34 Introduction to Maple Exercise 5.

What pattern do you see? Does this pattern continue forever? Exercise 6. Consider the sequence of odd counting numbers, grouped as shown below: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, ... 3 Notice that the first term is 1 , the sum of the next two terms is 8 or 23 , and the sum of the next three terms is 27 or 33 . The sum of the next four terms is 64 or 43 . Does this pattern continue? Use Maple to check that this pattern continues up to at least 153 . How could one be sure that it continues forever?

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