What are all values of $p$ for which $\displaystyle \int\limits_1^\infty \frac{1}{x^{3p+1}}  dx$ converges?

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Integrate using our knowledge of the power rule:

$$\int\limits_1^\infty \frac{1}{x^{3p+1}}  dx = \lim\limits_{b\to\infty} -\frac{1}{3p}\cdot\frac{1}{x^{3p}}\Big]_1^b$$

We can ignore the the coefficient since $p$ is a constant.

$$\lim_{b\to\infty} \frac{1}{b^{3p}}-\frac{1}{1^{3p}}$$

We can ignore the second term since it will evaluate to be a constant as well.

$$\lim_{b\to\infty} \frac{1}{b^{3p}}$$

If the exponent is negative, the integral will diverge regardless of the value of the exponent.

$$\lim_{b\to\infty} \frac{1}{b^{-a}} = \lim_{b\to\infty} b^a = \infty$$

If the exponent is positive, the denominator will approach infinity and the limit will evaluate to zero and therefore converge.

$$\lim_{b\to\infty} \frac{1}{b^{a}} = 0$$

For the exponent to be positive,

$$3p\gt 0$$ $$\boxed{p \gt 0}$$