Question

Does the following sequence converge or diverge? If it converges then find the limit. An = ln(n) / ln(n+1).

Answer 1

do you know l'Hospital's rule yet?

Answer 2

yes

Answer 3

taking the limit gives 0/0 so divergence test doesnt work

Answer 4

what is the derivative of the numerator?
what is the derivative of the denominator?

Answer 5

derivative of ln(n) = 1/n and derivative of ln(n+1) = 1/(n+1)

Answer 6

so if I take the limit of both they go to 0 right?

Answer 7

wait this is a sequence so I don't use any of the tests

Answer 8

so the sequence converges to 0?

Answer 9

they don't to zero, one is in the denominator

Answer 10

\[{\frac d{dn}\ln n\over\frac d{dn}\ln(n+1)}=?\]

Answer 11

ohh right I didnt flip the denominator x.x so it's n+1/n which means it diverges to infinity.

Answer 12

that does not diverge