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?

Concerning the ratio test, if you consider $$a_k=\ln\frac{k+5}{k+4}$$ then $$\frac{a_{k+1}}{a_k}=\frac{\log \left(\frac{k+6}{k+5}\right)}{\log \left(\frac{k+5}{k+4}\right)}=\frac{\log \left(1+\frac{1}{k+5}\right)}{\log \left(1+\frac{1}{k+4}\right)}$$ Now, use that, for small values of $y$, $$\log(1+y)=y-\frac{y^2}{2}+\frac{y^3}{3}+O\left(y^4\right)$$ Make  $y=\frac{1}{k+5}$ for the numerator, $y=\frac{1}{k+4}$ for the denominator, use the long division to arrive to $$\frac{a_{k+1}}{a_k}=1-\frac{1}{k}+\frac{11}{2 k^2}+O\left(\left(\frac{1}{k}\right)^3\right)$$
