132Lecture16.pdf - Lecture 16 Midterm Review 1 Know the...

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Lecture 16: Midterm Review. 1. Know the difference between sequence convergence and series conver- gence. Series a n converges Sequence { a n } converges, and lim a n = 0 Series a n diverges Can’t tell sequence { a n } Sequence { a n } converges Can’t tell series a n Sequence { a n } diverges or converges but lim a n 6 = 0 Series a n diverges lim b n 6 = 0 Alternating Series ( - 1) n b n , ( - 1) n +1 b n diverges Example: If a n = ln( n ) n , the sequence converges as lim n →∞ a n = 0 by L’Hospital Rule; but series a n diverges by integral or comparison test. 2. p-test. 1 n p converges ⇐⇒ p > 1 R 1 1 x p dx converges ⇐⇒ p > 1 3. Geometric series n =1 ar n - 1 converges to a 1 - r ⇐⇒ r < 1 Example: Given n =1 ( x - 1) n 8 n - 1 , this is a geometric series with r = x - 1 8 , a = x - 1 , the series converges when | r | < 1, so - 7 < x < 9, and it converges to a 1 - r = x - 1 1 - x - 1 8 . 4. Alternating series test (must verify (two) conditions first, only apply to alternating series).
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