Chapter 4(B) - Geometric Series For a 0, the series a + ar...

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Geometric Series ratio. (common) the is and first term the called is numbers, fixed are and where series, geometric a called is series the , 0 For 1 1 1 2 r a r a ar ar ar ar a a n n n = - - = + + + + + L L
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Geometric Series For this series, the -th partial sum is given by n ns n nn s r s a ar - =- 2 31 . n r s a r a r a r a r ar - =++ + ++ L 21 n n a r a r ar - =++ ++ L 1 1 n n r sa r - = - 1 r
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(i) 1 r = (ii) 1 r =- L + + + + a a a a L + - + - a a a a 21 n a a r a r ar - ++ + ++ LL Then if 0 (or if 0) n s n a aa = - ∞< Thus, the series is . divergent Thus, the series is . divergent Then { } is , 0, , 0, n s L
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(iii) If || 1, then 0. n rr <→ (iv) If 1, then (or ), and the series diverges. n -∞ 1 1 n n r sa r - = - Thus, . 1 n a s r - Hence, the sum of the series is . 1 a r -
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Convergence of Geometric Series 21 The geometric series with 0 converges to the sum if | | 1 1 and it diverges if 1. n a a r a r ar a a r r r - ++ + ++ < - LL
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Example 111 (i) is a geometric series 9 2 7 81 +++ L 11 first term and common ratio . 93 ar == 1 9 1 3 It converges to 1 =. 6 a r = --
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Example 11 (ii) 42 1 24 -+-+- L 2 4 44 221 4 1 1 2 8 3 a r  +-+- +=  -  =  --   = L 1 first term 4 and common ratio .
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Chapter 4(B) - Geometric Series For a 0, the series a + ar...

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