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assign5 - Assignment 5 Remarks and Partial Solutions 14.1...

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Assignment 5 Remarks and Partial Solutions 14.1 Determine which of the following series converge. Justify your answers. (a) n 4 2 n . (Hueristics: The exponential is the dominant term and will overcome any power. Therefore, we expect this to behave more like 1 2 n (although it never quite achieves this). For this combination of powers and exponential, it is equally advantageous to use the Ratio or Root Test.) lim sup n 4 2 n 1 /n = lim sup ( n 1 /n ) 4 2 ! = 1 2 . Therefore, it converges absolutely by the root test. (Note: The root test for the series 1 2 n also gives 1 2 .) (b) 2 n n ! . (Hueristics: You probably don’t have much intuition about the factorial function n !. It turns out that it behaves a little faster in growth than n n e - n by a factor like n (something called Stirling’s formula). You will see this when you apply the Ratio Test since the factor n n will drive the ratio to zero. Whenever you encounter the factorial function in a series, the ratio test is a natural first try.) lim sup a n +1 a n = lim sup 2 n + 1 = 0 . Therefore, it converges absolutely by the ratio test. (c) n 2 3 n . Similarly to (a), it converges absolutely by the root test. (d) n ! 3 n . Will diverge by the ratio test (see the hueristics of (b)). (e) cos 2 ( n ) n 2 . (Hueristics: The cosine function is bounded | cos( x ) | ≤ 1. Thus, one should see what else is there. In this case it is 1 n 2 which we know to sum nicely. Hence try to compare it to 1 n 2 .) Since a n = cos 2 ( n ) n 2 1 n 2 , the series converges by the comparison test.
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