I have a problem with a series. We have the series $$\sum \limits_{n=1}^{+\infty} \frac{1}{(n+2)(n+4)}.$$ When I used the integral test for convergence (correct me if I'm wrong), the result is $$\lim_{b \to +\infty} \left[ \frac{1}{2} \ln \left(\frac{b+2}{b+4}\right) - \frac{1}{2} \ln \frac{3}{5} \right]. $$ The limit will approach infinity (correct me if I'm wrong) or even an error as there is no $\ln 0$. However by comparison with $\frac{1}{n^2}$, it is convergent. Is there a mistake I made or did it really fail?
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$\frac{b+2}{b+4}$ tends to $1$ , so everything is alright. – Peter Jun 20 '20 at 15:40
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When $ b $ goes to infinity, the fraction $$\frac{b+2}{b+4}=\frac{1+\frac 2b}{1+\frac 4b}$$ goes to $ 1$ and its logarithm tends to zero. So, $$\int_1^{+\infty}\frac{dx}{(x+2)(x+4)}=-\frac 12\ln(\frac 35)>0$$
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