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*Here is some advanced study material for MA101 Calculus. This will be useful for students of KTU. Although it is not strictly the same syllabus of ktu it will give a thorough knowledge for you students*

*The purpose of this section is to discuss sums that contain infinitely many terms. The most familiar examples of such sums occur in the decimal representations of real numbers. For example, when we write 1 3 in the decimal form 1 3 = 0.3333 . . . , we mean 1 3 = 0.3 + 0.03 + 0.003 + 0.0003 + · · · which suggests that the decimal representation of 1 3 can be viewed as a sum of infinitely many real numbers. SUMS OF INFINITE SERIES Our first objective is to define what is meant by the “sum” of infinitely many real numbers. We begin with some terminology. 9.3.1 definition An infinite series is an expression that can be written in the form X` k=1 uk = u1 + u2 + u3 + · · · + uk + · · · The numbers u1, u2, u3, . . . are called the terms of the series. Since it is impossible to add infinitely many numbers together directly, sums of infinite series are defined and computed by an indirect limiting process. To motivate the basic idea, consider the decimal 0.3333 . . .*

sum of the series in an interval of length 0.001 (or less). Find the smallest value of n such that the interval containing π 4/90 in part (a) has a length of 0.001 or less. (c) Approximate π 4/90 to three decimal places using the midpoint of an interval of width at most 0.001 that contains the sum of the series. Use a calculating utility to confirm that your answer is within 0.0005 of π 4/90. 40. We showed in Section 9.3 that the harmonic seriesP` k=1 1/k diverges. Our objective in this problem is to demonstrate that although the partial sums of this series approach +`, they increase extremely slowly. (a) Use inequality (2) to show that for n ≥ 2 ln(n + 1) < sn < 1 + ln n (b) Use the inequalities in part (a) to find upper and lower bounds on the sum of the first million terms in the series. (c) Show that the sum of the first billion terms in the series is less than 22. (d) Find a value of n so that the sum of the first n terms is greater than 100. 41. Use a graphing utility to confirm that the integral test applies to the series P` k=1 k 2 e −k , and then determine whether the series converges. C 42. (a) Show that the hypotheses of the integral test are satisfied by the series P` k=1 1/(k3 + 1). (b) Use a CAS and the integral test to confirm that the series converges. (c) Construct a table of partial sums for n = 10, 20, 30, . . . , 100, showing at least six decimal places. (d) Based on your table, make a conjecture about the sum of the series to three decimal-place accuracy. (e) Use part (b) of Exercise 36 to check your conjecture

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