Math 182 Miniproject 6 Another $p$-test.md
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Math 182 Miniproject 6 Another $p$-test
===

**Overview:** In this project we develop a $p$-test to determine whether a certain type of integral converges or diverges.

**Prerequisites:** Section 6.5 of _Active Calculus_

In class we learned the $p$-test for integrals of the flavor
$$
\int_1^\infty\frac{1}{x^p}dx.
$$
__The $p$-test:__ $\int_1^\infty\frac{1}{x^p}dx$ converges if and only if $p>1$.

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Your task is to identify conditions on $p$ that let us know when the integral
$$
\int_2^\infty\frac{1}{x(\ln(x))^p}dx
$$
converges. You may want to break your exploration into separate cases. Include all of your work below.

$\int_{2}^{\infty\ }\frac{1}{x\left(\ln\left(x\right)\right)^{p}}dx$

$=\lim_{T \to \infty} \int_{2}^{T\ }\frac{1}{x\left(\ln\left(x\right)\right)^{p}}dx$

$=\int_{2}^{T}\left(\frac{1}{\ln\left(x\right)^{p}}\left(\frac{1} {x}\right)\right)dx$

let $u = ln(x)$
$du = (1/x)dx$

$=\lim_{T \to \infty}\int_{2}^{T}u^{-p}du$

$=\lim_{T \to \infty}\frac{1}{1-p}\left(\ln\left(x\right)\right)^{\left(1-p\right)}$ evaluated from 2 to T.

$=\lim_{T \to \infty}\left[\frac{1}{1-p}\left(\ln\left(T\right)^{\left(1-p\right)}\right)\right]-\left[\frac{1}{1-p}\ln\left(2\right)^{\left(1-p\right)}\right]$

Plugging in values for P where P is greater than one or P is less than or equal to one, we find that the function converges if P is greater than one and diverges if P is less than or equal to one.

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