Geometric Series Sequences and L Hopital s Rule soln - Mat104 Solutions to Problems from Old Exams Geometric Series Sequences and L'H^pital's Rule o(1

# Thomas' Calculus: Early Transcendentals

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Mat104 Solutions to Problems from Old Exams Geometric Series, Sequences and L’Hˆ opital’s Rule (1) Since e nx = ( e x ) n this is a geometric series with r = e x . It converges absolutely, provided | e x | < 1, that is for x ( -∞ , 0). In that case, it will converge to 1 1 - e x . (2) This is a geometric series with a = ( - 2 / 3) 4 and r = - 2 / 3. Therefore it converges to ( - 2 / 3) 4 1 + 2 / 3 = 16 135 . (3) Here we combine several geometric series: n =0 2 n 5 n = 1 1 - 2 / 5 = 5 / 3 n =0 3 n +1 5 n = 3 1 - 3 / 5 = 15 / 2 n =0 4 n +2 5 n = 16 1 - 4 / 5 = 80 the series we are given will converge to 5 / 3 + 15 / 2 + 80 = . . . . (4) Answer: 2 + 1 / 2 - 3 / 8 = 17 / 8 (similar to problems 1-3 above) (5) Answer: 8 / 3 + 2 = 14 / 3. (similar to problems 1-3 above) (6) As n → ∞ , both the numerator and the denominator go to infinity. Thus we can use L’Hˆ opital’s Rule: lim n →∞ ln( n 2 + n ) ln( n 2 - n ) = lim
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