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BC 3 Sequences and Series Quiz No Calculators allowed. Show set-up/method clearly on all problems. Name: n2 3n #1(4 pts) Given ak 2 for all n 1, determine whether the series 3n 2n 2 k 1 converges or diverges and if it converges, find its sum. n #2 (2 pts each) Suppose the series a n 1 n a k 1 k converges to 5 and an 0 for all n 1. Indicate whether each of the following statements is T (must be true) or F (false). You may write a quick one sentence explanation, this may get you partial credit. a. The sequence ak must converge. b. The partial sum S100 could have a value of 5.01. c. The sequence of partial sums must converge to 5. d. If the ratio test is applied to this series, then the value of lim ak 1 k ak must be a finite positive number strictly less than 1. e. If bk ak 0 for all k 1 , then b n 1 n must converge. #3(6 pts each) Determine whether each series converges or diverges. Explain reasoning carefully and completely. a. n! 3 n 1 b. n2 n 2 2n 3 2 n 5 3n n 5 c. n e n 1 n #3 (continued) Determine whether each series converges or diverges. Explain reasoning carefully and completely. d. 1 n 3 n 1 n n 1 approximates the value of the series k 1 3 k ! #4 (6 pts). Find a value of n such that S n k 1 with an error of at most .001. Explain carefully. You need only find the value of k! k 1 n, you need not find the approximation S n for this value of n. 3 k 1 , where f n is the Fibonacci sequence defined by: k 1 k f k 2 f1 f 2 1, and f n f n 1 f n 2 for n 3 . #5(3 pts) Evaluate f