# Calc6.20

Discuss the convergence or divergence of the series $\sum _{{n=1}}^{{\infty }}{\frac {\ln n}{n-3}}$.
We know the series $\sum _{{n=1}}^{{\infty }}{\frac {1}{n}}$ diverges because it is a p-series with p=1. Since
${\frac {1}{n}}<{\frac {1}{n-3}}<{\frac {\ln n}{n-3}}\,$
so, by the direct comparison, our series diverges. We can use the direct comparison test because our series has positive terms and so does the series $\sum _{{n=1}}^{{\infty }}{\frac {1}{n}}$. Again, this makes sense. If we have a series with positive terms which diverges, then it goes off to infinity. Another series with even bigger positive terms will clearly also go off to infinity.