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.
$$\int_2^\infty\frac{1}{x(\ln(x))^p}dx=$$
$$Lim_{B->∞}=\int_2^B\frac{1}{x(\ln(x))^p}dx$$
$$P=1; Lim_{B->\infty}=(Ln(B))-(Ln(2))=\infty(Diverges)$$
$$u=ln(x);du=1dx/x$$
$$Lim_{B->∞}\int_2^B\frac{du}{(u)^p}dx=Lim_{B->∞}\int_{Ln(2)}^{Ln(B)}\frac{du}{(u)^p}dx$$

$$\frac{u^{1-p}}{1-p}|^{lnB}_{ln2}=\frac{1}{1-p}Lim_{B->\infty}(LnB^{1-p}-Ln2^{1-p})$$

$$Converges$$
$$iff$$
$$p>1$$

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