TITLE: Series comparisons and logarithms?
QUESTION [6 upvotes]: Prove the convergence of
$$
\sum_{n = 1}^{\infty}
{\sqrt{\, 2n - 1\,}\,\ln\left(4n + 1\right) \over n\left(n+1\right)}
$$
I've been struggling for hours on this. By the textbook we have the limit test, comparison test, asymptotic test and integral test. When I use the asymptotic test my quotient ends up as $0$ or $\infty$, and I cannot prove anything. Since I am self studying, I have no one to turn to. The rest of the questions were easy, but I cant figure this out. Please help!

REPLY [2 votes]: $\ln n$ grows much, much, much more slowly than any positive power of $n$.  So for $n$ large enough,
$$ \frac{\sqrt{2n-1}\ln(4n+2)}{n(n+1)} < \frac{\sqrt{2n}*n^{1/10}}{n^2}<\frac{2}{n^{14/10}}.$$
Use the Comparison Test (then the Integral Test).  You may have to justify the first inequality, depending on who's asking.