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9.2LA-solutions - The sequence converges to 3 lim n →∞...

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MthSc 108-9.2 Sequences Learning Activity Solutions Show all work. For each sequence do the following: 1) Write out the first 5 terms as exact values; 2) Analytical determine if it converges or diverges; 3) If it converges state the value to which it converges; and 4) Make a concluding statement in a complete (nonmathematical )written sentence. 1. ln n 2 ( ) n = 0, ln4 2 , ln9 3 , ln16 4 , ln25 5 ,... The sequence converges to 0. Let f ( x ) = ln x 2 ( ) x so that we can apply L'Hospital's to evaluate the limit with the indeterminate form . lim x →∞ ln x 2 ( ) x = lim x →∞ 2ln x x = lim x →∞ 2 x = 0 So lim n →∞ ln n 2 ( ) n = lim x →∞ 2ln n n = 0. 2. sin n π 2 = 1,0, - 1,0,1,... { } The sequence diverges. The lim n →∞ sin n π 2 does not exist because it oscillates between values. 3. 3 n 2 n 2 + 1 = 3 2 , 12 5 , 27 10 , 48 17 , 75 26 ,...
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Unformatted text preview: The sequence converges to 3. lim n →∞ 3 n 2 n 2 + 1 = lim n →∞ 3 1 + 1 n 2 = 3 4. 3 n-1 4 n = 1 4 , 3 16 , 9 64 , 27 4 4 , 3 4 4 5 ,... The sequence converges to 0. Use the theorem lim n →∞ x n = 0 for x < 1 . Since 3 4 < 1, then lim n →∞ 1 4 3 4 n-1 = 1 4 lim n →∞ 3 4 n-1 = 1 4 ⋅ = . 5. 2 n + 1 ( ) ! 2 n-1 ( ) ! = 3! 1! , 5! 3! , 7! 5! , 9! 7! , 11! 9! ,... The sequence diverges. Simplify the factorials: 2 n + 1 ( ) ! 2 n-1 = 2 n + 1 ( ) 2 n ( ) 2 n-1 ( ) ! 2 n-1 = 2 n + 1 ( ) 2 n ( ) . Then ....
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