exercises_7 - n 2 n +1 + 3 n +1 ; (c) a n = p n 2 + n-p n...

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Analysis 1: Exercises 7 1. (a) Let a R . Let ( a n ) n N + be the sequence such that ( n N + ) ( a n = a ). Prove that lim n →∞ a n = a. (b) Let ( a n ) n N + be a sequence which diverges to and such that ( n N + ) ( a n 6 = 0). Prove that ± 1 a n ² n N + is a null sequence. 2. (a) Formulate in a single symbolic formula the negation to the statement: a R is the limit of the sequence ( a n ) n N + . (b) Prove that the following proposition is false: lim n →∞ 2 n n + 1 = 1. 3. Prove that the following sequences are null sequences. (a) a n = ( - 1) n +1 n ; (b) a n = 2 n n 3 + 1 ; (c) a n = 1 n ! ; (d) a n = n + 1 - n . 4. Let lim n →∞ a n = a . Prove that the following sequences converge and ﬁnd their limits (a) ( n N + ) ( b n = a n +1 - a n ); (b) ( n N + ) ( c n = | a n | ); (c) ( n N + ) ( d n = a n a n +1 ). 5. Let q R be such that | q | < 1. Prove that lim n →∞ q n = 0 . 6. Let lim n →∞ a n = a . Let b < a . Prove that a n > b for all but ﬁnitely many n N . 7. Compute the limits of the following sequences (a) a n = n 2 + 2 3 n 2 - 2 ; (b) a n = 2 n + 3

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Unformatted text preview: n 2 n +1 + 3 n +1 ; (c) a n = p n 2 + n-p n 2-n ; (d) a n = 1 1 2 + 1 2 3 + + 1 n ( n + 1) . 2 8. Prove the following equalities (a) lim n 2 n n ! = 0; (b) lim n n k a n = 0 ( a &gt; 1); (c) lim n a n n ! = 0; (d) lim n nq n = 0 , if | q | &lt; 1; (e) lim n n a = 1; (f) lim n n n = 1; (g) lim n 1 n n ! = 0 . 9. Let ( a n ) n N + be a convergent sequence of positive numbers. Prove that lim n n a 1 a 2 ...a n = lim n a n . 10. Let ( a n ) n N + be a sequence of positive numbers such that the limit lim n a n a n-1 exists. Prove that lim n n a n = lim n a n a n-1 ....
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This note was uploaded on 02/25/2010 for the course MATHEMATIC 11007 taught by Professor Spyros during the Winter '10 term at Bristol Community College.

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exercises_7 - n 2 n +1 + 3 n +1 ; (c) a n = p n 2 + n-p n...

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