SECTION 12.4 The Limit of a Sequence; Infinite Series 893 If <− > r r 1 or 1, then →∞ r lim n n does not exist, so the geometric series diverges. This is the same result that was obtained in Section 12.3. THEOREM Sum of a Geometric Series If − < < r 1 1, the geometric series ∑ = ∞ − ar k k 1 1 converges, and its sum is ∑ = − = ∞ − ar a r 1 k k 1 1 If ≤− ≥ r r 1 or 1, the geometric series ∑ = ∞ − ar k k 1 1 diverges. Concepts and Vocabulary 12.4 Assess Your Understanding 1. If a sequence { } sn converges to L, we call L the of the sequence. (a) limit (b) series (c) sum (d) term 2. If … … a a a , , , , n 1 2 is some collection of numbers, the expression ∑ = + + + + = ∞ a a a a k k n 1 1 2 is called a(n) . 3. If the sequence { } Sn of partial sums of an infinite series ∑ = ∞ a k k 1 has a limit S, then the series . 4. True or False An infinite series diverges if the sequence of partial sums diverges. Skill Building In Problems 5–28, determine whether each sequence converges or diverges. If it converges, find its limit. 5. { } { } = + a n n 4 2 1 n 6. { } { } = + a n n 5 2 1 n 7. b n n 5 4 3 n 2 { } { } = + 8. b n n 3 n 2 { } { } = + 9. { } { } = − + + b n n n 2 4 1 4 5 n 2 2 10. { } { } = + + + − c n n n n 4 2 7 8 2 n 3 3 2 11. { } { } = + a n n 5 1 n 2 12. { } { } = − + a n n n 12 3 1 n 2 3 13. { } { } = s 3 n n 14. { } { } = s 4 n n 15. { } ( ) { } = s 2 4 5 n n 16. { } ( ) { } = s 3 3 4 n n 17. { } { } = − b n 3 2 n 18. { } { } = + a n 10 1 n 19. { } { } = + − c n 4 ( 1) n n 20. { } { } ( ) = + − c n 1 1 n n 21. { } { } = + − a n n ( 1) n n 22. { } { } = + − b n n 2 ( 1) n n 2 2 23. { } { } = a n 2 n n 24. { } { } = a n e n n 25. π { ( )} { } = b n sin 2 n 26. π π ( ) { } { } = + s n cos 2 n 27. π { } { ( )} = s n cos n 28. π π ( ) { } { } = − a n sin 2 n In Problems 29–32, find the first five terms in the sequence of partial sums for each series. 29. ∑( ) = ∞ − 1 3 k k 1 1 30. ∑( ) = ∞ − 1 4 k k 1 1 31. ∑ = ∞ k k 1 32. ∑ = ∞ k k 1 2 1. Now Work 1. Modeling 1.ExplainingConcepts Calculus Preview 1.InteractiveFigure
RkJQdWJsaXNoZXIy NjM5ODQ=