Question

HELP QUIZ DUE! CONVERGE/DIVERGE/MONOTONE/BOUNDED? [1.] n/ln(n+1) [2.] (1/n)cos(1/n) ??

Answer 1

[3.] (2n^5)/(7n^3 +7)

Answer 2

with cosine series, you can use limit comparison test:if \[\lim_{n \rightarrow \infty}a _{n}/b _{n}=L>0\], then a series bexaves similarly like b series (if b converges, a converges and vice versa). so, if a=cos(1/n)/n, and b=n, \[\lim_{n \rightarrow \infty}\cos(1/n)/n ^{2}>1/n ^{2}\]. Series 1/n^2 converges, therefore converges \[\lim_{n \rightarrow \infty}\cos(1/n)/n ^{2}\]. So, we can apply limit comparison test. bn=x diverges, therefore diverges and (1/n)cos(1/n).

Answer 3

3 series diverges, because \[\lim_{n \rightarrow \infty}2n ^{5}/(7n ^{3}+1)=\lim_{n \rightarrow \infty}(2/7)*n ^{2}=infty\]

Answer 4

for the first, n/ln(n+1)<n/ln(n). The smaller series diverges (because n grows faster than ln(n)), therefore diverges and n/ln(n+1).

Answer 5

@mathmagician thanks heaps! how do you know if its monotone though?; if its bounded?; if x_n -> 0 as x -> infinity?

Answer 6

monotone series is series, where all terms are positive (or all terms are negative). Therefore cos series is not monotone in your case. The series is bounded, when you find a number, that is bigger than the sum of series' terms. Therefore all convergent series are bounded.

Answer 7

thanks a billion!