Determine if the following series converge or diverge. Show all work and explain

Question

Determine if the following series converge or diverge. Show all work and explain the reasoning for your determination (what test did you use, Root, Ratio..etc)

Explanation / Answer

c) Just use limit comparison with 1/n^(3/2). You'll get 1 as the limit so the series will converge. d) Use limit comparison with 1/n. You'll get 1 as the limit so the series diverges. e) Since ln(n) 3, then 1/ln(n) > 1/n. We know that the series of 1/n diverges, then the series 1/ln(n) diverges because it's greater than something that diverges (comparison test) f) Use Root Test. You'll get 2/3 as the limit, which is less than 1. So the series converges. g) Using Ratio Test, we get the lim of (2^(2(n+1)) / (2(n+1)+1)!) * ((2n+1)! / (2^(2n))) = (2^(2n+2)) / (2n+3)!) * ((2n+1)! / (2^(2n))) = 2^2 / ((2n+3)(2n+2)) = 0 Since that's less than 1, then the series converges.

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