Advanced Calculus fi..

Chapter 6 Infinite Series 385EXAMPLE 2The series, .Jis convergent. Here the partial sums form the sequence1which converges, as before, to 1. HenceFrom Theorem 5 of Section 6.4, one deduces a rule for addition and subtractionof convergent series, as well as a rule for multiplication by a constant:THEOREM 8tively and k is a constant, thenIf xzl a, and xzl b, are convergent with sums A and B respec-ifFor if one introduces the partial sums:A, =a, +a * . + a,,Sn = (a1 + 61) + . a . +B, =b, +...+b,,(a, +b,),then S, = A, + B,. Since A, converges to A and B, to B, S, converges to A + B.A similar reasoning holds for the series C(a, - b,), C ka,.The Cauchy criterion (Theorem 6) can also be interpreted in terms of series:THEOREM 9can be found such thatThe series Czl a, is convergent if and only if to each E > 0 an Nla,+l +a,+2+...+amI < E form > n > N. (6.16)This is a simple rewriting of condition (6.10). For, when m > n,6.6 TESTS FOR CONVERGENCE AND DIVERGENCE IA topic of prime importance is the formulation of rules.or "tests" that permit one todetermine whether a particular series converges or diverges. The problem is similarto and, in fact, closely related to that of improper integrals. This connection will bemade clear later.