calculus - How to determine whether $sum_{n=1}^{infty}lnleft(frac{n+2}{n+1}right)$ converges or diverges.

I am trying to find whether $\sum_{n=1}^{\infty}\ln\left(\frac{n+2}{n+1}\right)$ converges or diverges. I used the limit test, and it comes out as inconclusive since $\lim_{n\rightarrow\infty}\ln\left(\frac{n+2}{n+1}\right) = 0$. When I put it into wolfram, it states the series diverges by comparison test. But I don't know how to set up the comparison test (what series to compare it to). All help in solving this would be greatly appreciated, thanks.

Answer

Hints:

$\ln\frac{n+2}{n+1}=\ln(n+2)-\ln(n+1)$.

Then $$\sum_{n=1}^{\infty}\ln\frac{n+2}{n+1}=\lim_{m\to \infty}\sum_{n=1}^{m}\ln\frac{n+2}{n+1}=\lim_{m\to \infty} [\ln(m+2)-\ln2]=+\infty$$