Why doesn't the ratio test work to evaluate this series $$\sum_{k=1}^{\infty}\ln\frac{k+5}{k+4}$$
$$
\frac{\ln\frac{k+6}{k+5}}{\ln\frac{k+5}{k+4}}
=\cdots
$$
(ln(k+6/k+5)/ln(k+5/k+4))
ln((k+6/k+5)*(k+4/k+5))
simplify
lim as k approaches infinity of ln(kˆ2 +10k +24/ kˆ2 +10k+25)= ln(1) = 0.
why doesn't the series converge?

With equivalents (we can use equivalents because we have a series with positive termes): $$\ln \Bigl(\dfrac{k+5}{k+4}\Bigr)=\ln\Bigl(1+\dfrac 1{k+4}\Bigr)\sim_{+\infty} \dfrac 1{k+4}\sim_{+\infty} \dfrac 1k .$$
The series  has  a general term equivalent to the general term of the harmonic series, which diverges, hence it diverges.
