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.

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

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

For $p \not= 1$

$$\int_{2}^{\infty}\frac{1}{x\left(\ln\left(x\right)\right)}dx=lim_{T\to\infty}\int_{2}^{T}\frac{1}{u^{p}}du=lim_{T\to\infty}\int_{2}^{T}u^{-p}du=lim_{T\to\infty}\left[\frac{1}{1-p}x^{\left(1-p\right)}\right]_{2}^{T}$$

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

Which converges when $lim_{T\to\infty}\ln\left(T\right)^{1-p}=0$ if and only if $p>1$.

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