University:
Boston University
Course:
CAS MA 123 | Calculus I
Academic year:

2023
Views:

131

Pages:

2
Author:

Yung D

10.5 The Ratio and Root Tests 63 In "Ratio Test" Consider the infinite series {an with positive terms. Assume lim Antl an Then • if <1 then the series converges. • if exi then the series diverges. "or p = 00" • if p=1, then the test is inconclusive. Exp Apply Ratio test to | 2 en lim n300 an+1 an = lim (n+1)² e 5148 lim L e n (+)= Thus, the series converges by the ratio test. = 100 2 n! e lim antl an = lim (n+1)!.e n! lim = Thus, the series diverges by the ratio Test.. 100 =>1 3 Σ -1 "harmonic series which diverges" lim an+1 n→∞ an = lim =1 "Ratio Test is inclusive" "p-series with p=2 whic converges" lim ant! = an lim ۱ 2 2(144) 250 = lim 1=1 "Ratio 2 Test is inclusive" (1+) W (n+3)! converges by Ratio Test lion dan live ty → 3n+3 = <1 Th "The Rool Test" 64 Consider the infinite Series {an with an 20 for n> N. Assume Lim Van = p Then if <1 then the series converges • ife or infinit, then the series diverges • if p=1 then the test is inconclusive. Exp Apply the root test to " liman = lim "/(-)" = lim (-) η = ㅇ< Thus, the series converges by the root test. 1> 0= T my = lam mil= mo Thus, the series converges by the root test. lim Van = lim = lim и 124 (기)==3>) Thus, the series diverges by the root test. "harmonic series which diverges" Iman = lim = = 1 "Root Test lim p-series which converges poz" A is inconclusive lim Van = lim = lim 4 Test is inceandosive" Ip Consider the recursive defined termesi 9,=2, antian 92= Apply Ratio Test: 2 26 lim = lim 름 =0<1 Does an convery 9= 2 => The series converges. 4