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By Walter Thirring
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N=1 Mathematics for Physical Science and Engineering. 00002-X © 2014 Elsevier Inc. All rights reserved. 43 CHAPTER 2. INFINITE SERIES 44 It is important to be able to determine whether, or under what conditions, a series we would like to use is convergent. 1. The Geometric Series The geometric series, starting with u0 = 1 and with further terms formed by successive multiplications by a ﬁxed number r, is 1 + r + r2 + r3 + · · · + rn−1 + · · · . Its nth partial sum sn (that of the ﬁrst n terms) is sn = 1 − rn .
Determine whether each of the following series converges. ∞ (a) ∞ 1 n(n + 1) n=1 (b) ∞ (d) ∞ ln 1 + n=1 ∞ (c) (e) 1 n 1 1/n n · n n=1 (f) ∞ (h) arccot(n) n1/2 n=1 e−n n=0 ∞ ∞ 1 n ln n n=2 (g) 1 n2n n=1 (2n)! n! (3n)! n=0 ∞ (i) arccot(n) . 6. Try to evaluate the summations of the two preceding exercises using maple or mathematica. This approach will show that some of the divergent series are conﬁrmed as divergent and will give values of some of the convergent series. Notice that symbolic computation is not yet as accomplished as skilled humans in identifying series that are divergent.
The reader should make appropriate additions to his/her library as procedures of general utility are identiﬁed in later chapters. Ways of preserving and reusing coding are discussed later in this section. When dealing with arrays of data, there may be situations where it will be useful to understand how your symbolic computation system organizes the data and provides mechanisms for accessing or changing individual data items. A survey of this topic and some commands that may be useful for dealing with data structures can be found in Appendix C.