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

To begin identifying conditions on $p$ that les us know when the integral converges, we will solve the integral using limit. We apply the integral test to the series using the function $\int_{2}^{\infty}\frac{1}{x(ln(x))^p}$. In the special case $p=1$, we find the integral is $\int_{2}^{\infty}\frac{1}{x(ln(x))^p}dx$

$=\lim_{T \rightarrow \infty}\int_{2}^{T}\frac{du}{u^p}dx$

$=\lim_{T \rightarrow \infty}[ln(ln(T))-ln(ln(2))]$, which diverges.

If $p\neq 1$, we use u-substitution to get
$u=ln(x)$
$du=\frac{1}{x}dx$, this results in the following,

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

$=\int_{ln(2)}^{\infty}\frac{du}{u^p}$

$=\frac{u^{1-p}}{1-p}|_{ln(2)}^{\infty}$

$\frac{1}{1-p}\lim_{T \rightarrow \infty}[(T^{1-p})-(ln(2))^{1-p}]$.

If $p$ is chosen so that $\lim_{T \rightarrow \infty}T^{1-p}$ is convergent, then the integral converges as a result. If $p$ is chosen so that the limit diverges, then the integral will also. In conclusion, the conditions of $p$ that allow us to know when the integral converges is if and only if $p>1$.

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