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

Let us evaluate the integral.

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To use the Integral Test on the infinite series ( \sum_{n=1}^{\infty} \frac{1}{(2n+1)^3} ), you need to:

- Define the function ( f(x) = \frac{1}{(2x+1)^3} ).
- Check if ( f(x) ) is continuous, positive, and decreasing for ( x \geq 1 ).
- Integrate ( f(x) ) from 1 to ( \infty ).
- If the integral converges, then the series converges. If the integral diverges, then the series also diverges.

Now, integrating ( f(x) ):

[ \int_1^{\infty} \frac{1}{(2x+1)^3} , dx ]

[ \text{Let } u = 2x+1, , du = 2 , dx ]

[ \frac{1}{2} \int_{3}^{\infty} \frac{1}{u^3} , du ]

[ \frac{1}{2} \left[ -\frac{1}{2u^2} \right]_{3}^{\infty} ]

[ \frac{1}{4} \left( 0 - \left( -\frac{1}{18} \right) \right) ]

[ \frac{1}{72} ]

Since the integral converges, by the Integral Test, the series ( \sum_{n=1}^{\infty} \frac{1}{(2n+1)^3} ) also converges.

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