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

This preview shows pages 1–8. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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
(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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
(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 -
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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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 = --
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 .

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 21

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

This preview shows document pages 1 - 8. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online