Question

Does the following sequence converge or diverge? If it converges what is the limit? an=ln(n+1)-ln(n)

Answer 1

could write it as
\[a_n=\ln(\frac{n+1}{n})\] first and then check that one for convergence

Answer 2

Okay, I understand (and should have noticed) that ln(n+1)-ln(n)=ln((n+1)/n). Now, the limit a n approaches inf for ln((n+1)/n) = inf/inf. Does this then equal 1, or is this an indeterminate that then needs manipulation by L'Hopital's Rule?

Answer 3

\[\lim_{n\to\infty}\frac{n+1}{n}=\lim_{n\to\infty}1+\frac{1}{n}=1+0=1\]

\[\ln(1)=0\]