\(\displaystyle\sum_{n=1}^\infty\frac2{(2n+1)^3}<\int_1^\infty\frac2{(2x-1)^3}\,\mathrm dx\)
If the integral converges, then so will the sum.
Let \(y=2x-1\), so that \(\mathrm dy=2\,\mathrm dx\). Then the integral is
\(\displaystyle\int_1^\infty\frac{\mathrm dy}{y^3}=\lim_{c\to\infty}\int_1^c\frac{\mathrm dy}{y^3}\)
\(\displaystyle=\lim_{c\to\infty}-\frac1{2y^2}\bigg|_{y=1}^{y=c}\)
\(\displaystyle=\lim_{c\to\infty}-\frac1{2c^2}+\frac1{2\times1^2}\)
\(=\dfrac12\)
so the sum converges by the integral test (to some number less than 1/2).
