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Calculus: An Integrated Approach to Functions and their Rates of Change, Preliminary Edition Part 116

Calculus: An Integrated Approach to Functions and their Rates of Change, Preliminary Edition Part 116. A major complaint of professors teaching calculus is that students don't have the appropriate background to work through the calculus course successfully. This text is targeted directly at this underprepared audience. This is a single-variable (2-semester) calculus text that incorporates a conceptual re-introduction to key precalculus ideas throughout the exposition as appropriate. This is the ideal resource for those schools dealing with poorly prepared students or for schools introducing a slower paced, integrated precalculus/calculus course | Proofs to Accompany Chapter 30 Series 1 1 3 1 n lim n lim a 1 3 x Jx a 1 3 A TO The sequence of partial sums is bounded and increasing. Therefore by the Bounded Increasing Partial Sums Theorem TO 1 a converges. Suppose f x Jx to. Refer to Equation H.1 to obtain n 1 x Jx a1 3 a2 3-3 an n Taking the limit as n to gives to lim n. n TO So n grows without bound and TO1 a diverges. Now we show that the behavior of the integral can be determined by that of the series. Because x 0 decreasing and continuous on 1 to lim6 TO f6 x Jx is either flnite or grows without bound. Therefore if we can flnd an upper bound the integral converges. If it has no upper bound it diverges. Suppose TO 1 a converges. Denote its sum by . From Equation H.1 we know n J x Jx a1 3 a2 3-3 an 1 n n 1 lim x Jx lim a . nn 1 1 If lim TO n x Jx is bounded so too is lim6 TO f6 x Jx given the hypotheses . Suppose TO 1 a diverges. Because the terms are all positive we know limn TO 1 a to. From Equation H.1 we know n a2 3---3 an J x Jx nn lim a lim x Jx. nn 2 We conclude that lim6 TO f6 x Jx to the improper integral diverges. Abel Niels 382 1078 Absolute convergence conditional 953 explanation of 952-953 implies convergence 1128-1129 Absolute maximum point 347 Absolute maximum value 252 347 Absolute minimum point 347 Absolute minimum value 347 352 Absolute value function explanation of 61-62 maximum and minimum and 350 Absolute values analytic principle for working with 66 elements of 65 functions and 129 geometric principle for working with 66-68 Acceleration due to force of gravity 150 406-407 explanation of 76 Accruement 746 Accumulation 746 Addition of functions 101-103 principles of 1059-1061 Addition formulas 668-669 709 Additive integrand property 738 Algebra equations and 1053 explanation of 1051-1052 exponential 247 309-312 exponents and 1053-1054 expressions and 1052-1053 1056-1070 order of operations and 1054 solving equations using 1071-1083 See also Equations square roots and 1054-1055 .