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Unformatted text preview: Proposition 3 The geometric series summationdisplay m =0 q m = 1 + q + q 2 + (783) converges with the sum 1 / (1 q ) if | q | < 1 and diverges if | q | 1 . Proof: For | q | 1 we have | q m | 1, m 0, and Thm. 14 implies divergence. Let | q | < 1. The n-th partial sum is given by s n = 1 + q + + q n , n N , (784) so that qs n = q + + q n + q n +1 . (785) After subtraction, we are left with s n qs n = (1 q ) s n = 1 q n +1 . (786) Since q negationslash = 1, we have 1 q negationslash = 0, and we may solve (786) for s n : s n = 1 q n +1 1 q = 1 1 q q n +1 1 q 1 1 q , n , (787) since | q | < 1. square In the following sections, we will mostly use the ratio test to establish convergence of a series. Theorem 17 If a series z 1 + z 2 + with z n negationslash = 0 , n N , has the property that for some N N and for some q < 1 vextendsingle vextendsingle vextendsingle vextendsingle z n +1 z n vextendsingle vextendsingle vextendsingle vextendsingle q, n > N, (788) then this series converges absolutely. If vextendsingle vextendsingle vextendsingle vextendsingle z n +1 z n vextendsingle vextendsingle vextendsingle vextendsingle 1 , n > N, (789) then the series z 1 + z 2 + diverges. 134 Remark: q = 1 is not sufficient (cf. harmonic series)! Proof: From (789), we have that | z n +1 | | z n | , n > N , so divergence of the series follows from Thm. 14. If (788) holds, then | z n +1 | q | z n | , n > N . In particular, | z N + p | q | z N + p- 1 | q 2 | z N + p- 2 | q p- 1 | z N +1 | , p N . With Prop. 3, we find that | z N +1 | + | z N +2 | + | z N +1 | ( 1 + q + q 2 + ) | z N +1 | 1 1 q . (790) The comparison test (Thm. 16) yields absolute convergence of the series z 1 + z 2 + . square If the sequence of ratios in Thm. 17 converges, we get the more convenient Theorem 18 If a series z 1 + z 2 + with z n negationslash = 0 , n N , is such that lim n vextendsingle vextendsingle vextendsingle vextendsingle z n +1 z n vextendsingle vextendsingle vextendsingle vextendsingle = L, (791) then a) If L < 1 , the series converges absolutely. b) If L > 1 , the series diverges. c) If L = 1 , the series may converge or diverge, i. e. the test fails and permits no conclusion. Proof: a) We define k n := | z n +1 /z n | and let L = 1 b < 1. By definition of the limit, k n must eventually get close to 1 b , say, k n q := 1 b/ 2 < 1, n > N , for some N . Convergence of z 1 + z 2 + now follows from Thm. 17. b) For L = 1 + c > 1 we have k n 1 + c/ 2 > 1, n > N * , for some N * sufficiently largte, which implies divergence of z 1 + z 2 + by Thm. 17....
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