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.

Case 1:

If $p=1$ we get $\int_2^\infty\frac{1}{x(\ln(x))}dx$.
$=\lim_{T \to \infty} \int_2^T\frac{1}{x(\ln(x))}dx$ Then we will do u-substitution where $u=ln(x)$ and $du=\frac{1}{x}dx$.
$=\lim_{T \to \infty}\int_2^T\frac{1}{u}dx$ [The bounds change to ln(2) and ln(T) but I'm have trouble changing it neatly. I tried looking it up as well.]
$=\lim_{T \to \infty} ln|u|$ Bounds from $ln(2)$ to $ln(T)$.
$=\lim_{T \to \infty} ln|ln(T)| -ln|ln(2)|$
$= \infty$

Case 2:

If $p>1$ we get $\int_2^\infty\frac{1}{x(\ln(x))^p}dx$
$=\lim_{T \to \infty} \int_2^T\frac{ln(x)^-p}{x}dx$. Then we use u-substitution where $u=ln(x)$ and $du=\frac{1}{x}dx$
$=\lim_{T \to \infty} \int_2^Tu^-p du$ The new bounds would be $ln(2)$ to $ln(T)$.
$=\lim_{T \to \infty} \frac{1}{1-p} (u^(1-p))$ Bounds from $ln(2)$ to $ln(T)$.
$=\lim_{T \to \infty} \frac{1}{1-p}ln(T)^(1-p))-\frac{1}{1-p}(ln(2)^(1-p))$
$= 0 -\frac{1}{1-p}(ln(2)^(1-p)))$

$\int_2^\infty\frac{1}{x(\ln(x))^p}dx$ converges if and only if $p>1$

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