# Math Help - integral test convergence divergence

1. ## integral test convergence divergence

im confused on this one

$\sum_{n = 2}^{\infty} 6n^2 \exp(-n^3)$

i think the exponential goes to 0 does this mean that i can not use the integral test?

2. Given $a_n=6n^2e^{-n^3}.$ This is clearly positive and continuous, but is it a strictly decreasing sequence? This is the last condition so that you can apply the integral test.

3. Originally Posted by acosta0809
im confused on this one

$\sum_{n = 2}^{\infty} 6n^2 \exp(-n^3)$

i think the exponential goes to 0 does this mean that i can not use the integral test?
If you need solution

$6\int\limits_2^\infty {{x^2}{e^{ - {x^3}}}dx} = - 2\int\limits_2^\infty {{e^{ - {x^3}}}d\left( { - {x^3}} \right)} = \left. { - 2{e^{ - {x^3}}}} \right|_2^\infty = - 2 \cdot \underbrace {\mathop {\lim }\limits_{x \to \infty } \frac{1}{{{e^{{x^3}}}}}}_0 + \frac{2}{{{e^{{2^3}}}}} = \frac{2}{{{e^8}}}.$