Convergence of series $\sum_{n=1}^\infty \ln\left(\frac{2n+7}{2n+1}\right)$? I have the series
$$\sum\limits_{n=1}^\infty \ln\left(\frac{2n+7}{2n+1}\right)$$
I'm trying to find if the sequence converges and if so, find its sum.
I have done the ratio and root test but It seems it is inconclusive.
How can I find if the series converges? 

No need for convergence tests! Note that if $f(n)=\ln\left(\dfrac{2n+7}{2n+1}\right)$ then:
$$f(n)+f(n+3)=\ln (2n+13)-\ln (2n+1)$$
So most terms cancel out. In other words, the partial sum of your series is:
$$\begin{align*}\sum_1^m f(n)&=-\ln 3-\ln 5-\ln 7+\ln(2m+3)+\ln(2m+5)+\ln(2m+7)\\
&= \ln(2m+3)(2m+5)(2m+7)-\ln 105\end{align*}$$
Which obviously does not converge.
