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A B C D A C B D A D B C . 10.87. (a) Prove that r a sin u cos v i a sin u sin v j a cos u, 0 @ u @ ; 0 @ v < 2 represents a sphere of radius a. (b) Use Problem 10.86 to show that the surface area of this sphere is 4a2 . 10.88. Use the result of Problem 10.34 to obtain div A in (a) cylindrical and (b) spherical coordinates. See Page 161. In nite Series
The early developers of the calculus, including Newton and Leibniz, were well aware of the importance of in nite series. The values of many functions such as sine and cosine were geometrically obtainable only in special cases. In nite series provided a way of developing extensive tables of values for them. This chapter begins with a statement of what is meant by in nite series, then the question of when these sums can be assigned values is addressed. Much information can be obtained by exploring in nite sums of constant terms; however, the eventual objective in analysis is to introduce series that depend on variables. This presents the possibility of representing functions by series. Afterward, the question of how continuity, di erentiability, and integrability play a role can be examined. The question of dividing a line segment into in nitesimal parts has stimulated the imaginations of philosophers for a very long time. In a corruption of a paradox introduce by Zeno of Elea (in the fth century B.C.) a dimensionless frog sits on the end of a onedimensional log of unit length. The frog jumps halfway, and then halfway and halfway ad in nitum. The question is whether the frog ever reaches the other end. Mathematically, an unending sum, 1 1 1 n 2 4 2 is suggested. Common sense tells us that the sum must approach one even though that value is never attained. We can form sequences of partial sums 1 1 1 1 1 1 S1 ; S2 ; . . . ; Sn n 2 2 4 2 4 2 and then examine the limit. This returns us to 2 and the modern manner of thinking about the in nitesimal. In this chapter consideration of such sums launches us on the road to the theory of in nite series. DEFINITIONS OF INFINITE SERIES AND THEIR CONVERGENCE AND DIVERGENCE De nition: The sum
1 X n 1
un u1 u2 un 265
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INFINITE SERIES
[CHAP. 11
is an in nite series. Its value, if one exists, is the limit of the sequence of partial sums fSn g S lim Sn If there is a unique value, the series is said to converge to that sum, S. If there is not a unique sum the series is said to diverge. 1 X1 Sometimes the character of a series is obvious. For example, the series generated by the 2n 1 X n 1 n is divergent. On the other hand, the variable series frog on the log surely converges, while 1 x x 2 x3 x4 x5 raises questions. 1 . If 1 < x < 1, the sums Sn yields an This series may be obtained by carrying out the division 1 x 1 and (2) is the exact value. The indecision arises for x 1. Some very great approximations to 1 x mathematicians, including Leonard Euler, thought that S should be equal to 1, as is obtained by 2 1 . The problem with this conclusion arises with examination of substituting 1 into 1 x 1 1 1 1 1 1 and observation that appropriate associations can produce values of 1 or 0. Imposition of the condition of uniqueness for convergence put this series in the category of divergent and eliminated such possibility of ambiguity in other cases. FUNDAMENTAL FACTS CONCERNING INFINITE SERIES 1. If un converges, then lim un 0 (see Problem 2.26, Chap. 2). The converse, however, is not necessarily true, i.e., if lim un 0, un may or may not converge. n!1 n!1

