Multivariable Calculus: Concepts and Contexts by James Stewart

By James Stewart

Stewart's MULTIVARIABLE CALCULUS: suggestions AND CONTEXTS, 3rd variation bargains a streamlined method of instructing calculus, targeting significant techniques and aiding people with detailed definitions, sufferer causes, and thoroughly graded difficulties. MULTIVARIABLE CALCULUS: ideas AND CONTEXTS is very seemed since it has effectively introduced peace to departments that have been cut up among reform and standard methods to educating calculus. not just does the textual content support reconcile the 2 colleges of suggestion by way of skillfully merging the easiest of conventional calculus with the easiest of the reform move, it does so with innovation and meticulous accuracy.

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N SOLUTION The function f ͑x͒ ෇ ͑ln x͒͞x is positive and continuous for x Ͼ 1 because the logarithm function is continuous. But it is not obvious whether or not f is decreasing, so we compute its derivative: f Ј͑x͒ ෇ x͑1͞x͒ Ϫ ln x 1 Ϫ ln x ෇ 2 x x2 Thus, f Ј͑x͒ Ͻ 0 when ln x Ͼ 1, that is, x Ͼ e. It follows that f is decreasing when x Ͼ e and so we can apply the Integral Test: y ϱ 1 ln x t ln x ͑ln x͒2 dx ෇ lim y dx ෇ lim tlϱ 1 tlϱ x x 2 ͬ t 1 ͑ln t͒ ෇ϱ 2 2 ෇ lim tlϱ Since this improper integral is divergent, the series ͸ ͑ln n͒͞n is also divergent by the Integral Test.

See Exercises 46 and 47 for two geometric interpretations of Example 6. 1 2 1 1 Ϫ 2 3 ϩ 1 nϩ1 lim sn ෇ lim nlϱ nlϱ 1 1 Ϫ 3 4 ϩ ͩ 1Ϫ 1 nϩ1 ͪ ϩ иии ϩ ͩ 1 1 Ϫ n nϩ1 ͪ ෇1Ϫ0෇1 ■ ■ Therefore, the given series is convergent and ϱ ͚ n෇1 1 ෇1 n͑n ϩ 1͒ EXAMPLE 7 Show that the harmonic series ϱ 1 ͚ ͕sn ͖ n෇1 1 1 1 1 ෇ 1 ϩ ϩ ϩ ϩ иии n 2 3 4 is divergent. SOLUTION For this particular series it’s convenient to consider the partial sums s2 , s4 , s8 , s16 , s32 , . . and show that they become large. ͕a n ͖ 0 s2 ෇ 1 ϩ 12 n s4 ෇ 1 ϩ 2 ϩ ( 3 ϩ 4 ) Ͼ 1 ϩ 2 ϩ ( 4 ϩ 4 ) ෇ 1 ϩ 2 1 FIGURE 3 1 1 1 1 1 2 s8 ෇ 1 ϩ 12 ϩ ( 13 ϩ 14 ) ϩ ( 15 ϩ 16 ϩ 17 ϩ 18 ) Ͼ 1 ϩ 2 ϩ (4 ϩ 4) ϩ (8 ϩ 8 ϩ 8 ϩ 8) 1 1 1 1 1 1 1 ෇ 1 ϩ 12 ϩ 12 ϩ 12 ෇ 1 ϩ 32 s16 ෇ 1 ϩ 12 ϩ ( 13 ϩ 14 ) ϩ ( 15 ϩ и и и ϩ 18 ) ϩ ( 19 ϩ и и и ϩ 161 ) Ͼ 1 ϩ 12 ϩ ( 14 ϩ 14 ) ϩ ( 18 ϩ и и и ϩ 18 ) ϩ ( 161 ϩ и и и ϩ 161 ) 1 1 1 1 4 ෇1ϩ2ϩ2ϩ2ϩ2෇1ϩ2 572 ■ CHAPTER 8 INFINITE SEQUENCES AND SERIES Similarly, s32 Ͼ 1 ϩ 52 , s64 Ͼ 1 ϩ 62 , and in general s2 n Ͼ 1 ϩ The method used in Example 7 for showing that the harmonic series diverges is due to the French scholar Nicole Oresme (1323–1382).

Or will it exhibit random behavior? Write a program to compute the first n terms of this sequence starting with an initial population p0 , where 0 Ͻ p0 Ͻ 1. Use this program to do the following. 1. Calculate 20 or 30 terms of the sequence for p0 ෇ 2 and for two values of k such that 1 1 Ͻ k Ͻ 3. Graph the sequences. Do they appear to converge? Repeat for a different value of p0 between 0 and 1. Does the limit depend on the choice of p0? Does it depend on the choice of k ? 2. 4 and plot them. What do you notice about the behavior of the terms?

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