Question

How do you use the Integral test on the infinite series `sum_(n=1)^oo1/(2n+1)^3` ?

Answer 1

Since the corresponding integral
`int_1^infty 1/(2x+1)^3dx` converges to `1/36`,
the series
`sum_{n=1}^infty1/(2n+1)^3` also converges by Integral Test.

Let us evaluate the integral.

`int_1^infty 1/(2x+1)^3dx=int_1^infty(2x+1)^{-3}dx`

Let `u=2x+1`. #Rightarrow {du}/{dx}=2 Rightarrow dx={du}/2#
`x: 1 to infty Rightarrow u: 3 to infty`

`=int_3^infty u^{-3}{du}/2`

`=1/2lim_{t to infty}int_3^t u^{-3}du`

`=1/2lim_{t to infty}[u^{-2}/{-2}]_3^t`

`=-1/4lim_{t to infty}(1/t^2-1/3^2)`

`=-1/4(0-1/9)`

`=1/36`