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Unformatted text preview: MAC 2312 Quiz 5 February 21, 2012 SOLUTIONS
1. Determine whether the sequence converges or diverges. If the sequence converges, compute its limit. n2 + 2n + 1 an = n = 1, 2, 3, 3n2  2 Solution: Use the related function f (x) = x2 + 2x + 1 . 3x2  2 Notice that an = f (n) for each n = 1, 2, 3, . Moreover, since the degree of f 's numerator and the degree of f 's denominator coincide, 1 lim f (x) = . x 3 We therefore conclude that 1 lim an = . n 3 2. In this problem, you will use a geometric series to show that 9.9 = 10. Recall that 9.9 = 9.9999 (infinitely repeating 9 s). This may be rewritten as 9.9 = 9 + 9 9 9 9 + + + + 10 100 1000 10, 000 1 1 1 1 + + + + = 9 1+ 10 100 1000 10, 000 . (1) (a) Find the value of r such that 1 + Solution: 1+ 1 1 1 + + + = 10 100 1000 = 1 10 + 1 10 1 100 + 1 1000 + =
1 n n=0 r . 0 + 1 10
n 1 10 + 1 10 2 + 1 10 3 + n=0 That is, r = 1 . 10 (b) Use the formula for the sum of a geometric series along with the value of r you 1 1 1 found in part (a) to compute the value of the sum 1 + 10 + 100 + 1000 + . Solution: The sum for a geometric series is given by rn =
n=0 1 1r
1 10 for r < 1. (2) Using part (a) and plugging r = into equation (2), we get 1 1 1 + + + = 1+ 10 100 1000 = n=0 1 10 n 1 1 1  10 10 = 9 (c) Plug the result you obtained in part (b) into the righthand side of equation (1) and simplify to conclude that 9.9 = 10. Solution: Using the result of part (b) in equation (1) we obtain 9.9 = 9 1 + = 9 = 10. 10 9 1 1 1 1 + + + + 10 100 1000 10, 000 . ...
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This note was uploaded on 03/27/2012 for the course MAC 2312 taught by Professor Bonner during the Spring '08 term at University of Florida.
 Spring '08
 Bonner
 Calculus

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