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3100exam2 - MATH 3100 SECOND MIDTERM EXAM WITH SOLUTIONS...

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MATH 3100 SECOND MIDTERM EXAM: WITH SOLUTIONS Directions: Please solve all the problems. Calculators are not permitted. You have 55 minutes. Good luck! 1) a) State the nth term test for convergence. Solution: Let n a n be a convergent real series. Then lim n →∞ a n = 0. (Or in contrapositive form: if a n 9 0, then the series n a n diverges.) b) Prove the n th term test for convergence. Solution: Let A n = a 1 + . . . + a n . Our hypothesis is that A n A , say. But a n = A n - A n - 1 , so lim n →∞ a n = lim n →∞ A n - lim n →∞ A n - 1 = A - A = 0 . Alternately, this is a very special case of the Cauchy Criterion for convergence. c) If for a sequence { a n } we have lim n →∞ a n = 0, must n a n converge? Ei- ther prove this or give a counterexample. Solution: This is false, the most famous counterexample being the harmonic se- ries n 1 n : 1 n 0 but the series diverges according to either the Condensation Test or the Integral Test. (Comment: if instead of asking simply that the n th term approach zero, we required that all the tails of the series get arbitrarily small, then we do get a criterion for convergence which is both necessary and sufficient, a restatement of Cauchy’s Criterion. But this is of mostly theoretical use...) 2) Let r 0. Consider a series n =1 a n with a n 0 for all n . Suppose moreover that for all n 10, a n = 17 r n .
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