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# finalsol - Math 9C Final exam Study guide C Pro Fall 2011...

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Math 9C Final exam Study guide C. Pro - Fall 2011 The following document consists of the first two study guides and a new study guide in section 3. 1 Concept questions What is the difference between a sequence and a series? There is no difference, a sequence is a list of numbers, a series is also sequence (a sequnceof partial sums) Can you give the definition of what it means for a sequence to converge? A sequence { a n } converges if for every > 0, there is a number L and an integer N such that for every k > N , then | a k - L | < What is the definition for a series to converge? A series converges if its sequence of partial sums converges. If the a k ’s are nonnegative, why does this say the sequence of partial sums is an increasing sequence? If { s n } is the sequence of partial sums, then s n +1 - s n = a n +1 0 and so the s n s are increasing. What is the Monotone bounded convergence theorem? A sequence { a n } that is monotone and bounded must converge. Can you explain how the integral test is used to show your sequence of partial sums is bounded? The area under the curve of f ( x ) is larger than the sum of the rectangles with area a k where f ( k ) = a k . If R 1 f ( x ) dx converges to some nubmer M , this means that the s n ’s are bounded above by this number. How did we come up with the p -test? We used the integral test with the function f ( x ) = 1 x p How are the integral test and the comparison test similar? Both determine conver- gence or divergence of a series a k by knowledge of the convergence or divergence of something else. In the case of the integral test, the convergence or divergence of an improper integral and in the case of the comparison test, the convergence or divergence of another series. 1

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Math 9C Final exam Study guide C. Pro - Fall 2011 2 Sample Problems State the integral test, comparison test, limit comparison test, alternating series test, ratio test, and root test. See the notes. Find the n -th term of the sequence { 1 , 0 , 1 , 0 , 1 , 0 , . . . } a n = 1 2 + 1 2 ( - 1) n n = 0 , 1 , 2 , 3 , 4 , . . . Find the limits as n → ∞ of the following sequences • { 3 1 /n } lim n →∞ 3 1 n = 3 lim n →∞ 1 n = 3 0 = 1 • { 4 n 2 +2 n 3 +3 } lim n →∞ 4 n 2 + 2 n 3 + 3 = 0 Power of n in denom. is larger than num. • { 1 2 n } lim n →∞ 1 2 n = 0 1 2 < 1 • { n n } Hint: n n = e ln( n ) n (Do you know why?) lim n →∞ n n = lim n →∞ e ln( n ) n = e lim n →∞ ln( n ) n = e lim n →∞ 1 n L’Hospitals rule = e 0 = 1 • { n sin 1 n } lim n →∞ n sin 1 n = lim n →∞ sin 1 n 1 n = 1 lim θ →∞ sin θ θ = 1 2
Math 9C Final exam Study guide C. Pro - Fall 2011 • { n ln ( 1 + 1 n ) } Hint: L’Hospital’s rule. lim n →∞ n ln 1 + 1 n = ln ( 1 + 1 n ) 1 n = lim n →∞ 1 ( 1+ 1 n ) ( 1 + 1 n ) 0 ( 1 n ) 0 = lim n →∞ 1 ( 1 + 1 n ) = 1 • { ( 1 + 1 n ) n } Hint: ( 1 + 1 n ) n = e n ln ( 1+ 1 n ) (Do you know why?) lim n →∞ 1 + 1 n n = lim n →∞ e n ln ( 1+ 1 n ) = e lim n →∞ n ln ( 1+ 1 n ) = e 1 = e Do the following series converge or diverge?

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finalsol - Math 9C Final exam Study guide C Pro Fall 2011...

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