Test 2b solutions

# Test 2b solutions - Solution to the Second Test C1 I a True...

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Solution to the Second Test C1 I: a) True, b) False, f (0) = f (1) = 0 c) False (Example a n = 1 /n , b n = n .), d) 32, e) ln(exp(ln(9))) + ln(exp(ln(1 / 9))) = ln(9) + ln(1 / 9) = ln(1) = 0. II: a) 4, b) does not exist, c) 3, d) (1 - 1 /n - 6 /n 2 ) = (1 - 3 /n )(1 + 2 /n ) and hence lim n →∞ (1 - 1 /n - 6 /n 2 ) n = lim n →∞ (1 - 3 /n ) n lim n →∞ (1 + 2 /n ) n = e - 3 e 2 = e - 1 . e) If we set k = n the expression is the same as lim k →∞ (1 + 1 k ) k 2 and as a ±rst guess for k large (1 + 1 k ) k 2 e k and hence one expects that the limit does not exist. This is a good argument. Another one is to use Bernoulli’s inequality (1 + 1 k ) k 2 1 + k 2 1 k = 1 + k which also shows that the limit does not exist. III: a) Clarly 0 a n 1 since ( n + 1) / (10 n ) 1 and a n = 1. b) The sequence is decreasing. c) The sequence is decreasing and bounded below by zero and hence it converges. d)

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## This note was uploaded on 02/07/2011 for the course MATH 1501 taught by Professor N/a during the Fall '08 term at Georgia Tech.

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Test 2b solutions - Solution to the Second Test C1 I a True...

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