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Unformatted text preview: Practice Exam 1 9/22/06 Name: No calculators allowed. Show all work. DO NOT BEGIN TEST UNTIL INSTRUCTED TO DO SO. Write out and sign the Georgia Tech Honor Challenge below. I commit to uphold the ideals of honor and integrity by refusing to betray the trust bestowed upon me as a member of the Georgia Tech community. 1 2 (1) (a) Give the mathematical definition of the statement the sequence ( a n ) converges to the limit L . The sequence ( a n ) converges to the limit L if for any > 0 there exists an N > 0 such that | a n- L | < for every n > N . (b) Explain the meaning of this definition in common English. The definition says that given any distance, from some term onwards the sequence will always be less than that distance away from the limit. Alternatively, the sequence will eventually remain closer to the limit than any specified distance. (2) (a) Show that 7 n 2 = O ( n 2- 1). We may simply show that the sequence | 7 n 2 n 2- 1 | = 7 n 2 n 2- 1 converges. We have that lim n 7 n 2 n 2- 1 = lim n 7 1- 1 n 2 = 7 1- = 7 where the second-to-last equality is due to the facts that the function 7 1- x is continuous at x = 0 and that lim n 1 n 2 = 0. Therefore 7 n 2 n 2- 1 converges and hence 7 n 2 = O ( n 2- 1). (b) Show that sin( n ) = O ( 1 2 ). For the definition of big Oh we must show that there are constants C and N such that | sin( n ) | C | 1 2 | for each n N . Since | sin( n ) | 1 for all n we may just choose C = 2 and use any N . That is to say | sin( n ) | 1 = 2 | 1 2 | for all n ....
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- Fall '09