Question

Use the Integral Test to determine whether the series is convergent or divergent. (If the quantity diverges, enter DIVERGES.)

Answer 1

$\displaystyle\sum_{n=1}^\infty\frac2{(2n+1)^3}<\int_1^\infty\frac2{(2x-1)^3}\,\mathrm dx$

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).