Another Math.SE Double Integral Using Polar Coordinate

I wanted to post the following answer to a question on double integral on Math.SE, but someone had submitted his work before I finished typing. As a result, I’m posting this on my personal blog. Let $r = \sqrt{x^2+4y^2}$ and $t = \begin{cases} \tan^{-1}(2y/x) &\text{ if } x > 0 \\ \pi/2 &\text{ if } x = 0. \end{cases}$ Then $\begin{cases} x &= r \cos t \\ y &= (r \sin t)/2 \end{cases}$ and $D = { (r,t) \mid r \ge 0, t \in [\pi/4, \pi/2] }$. [Read More]

Solution to a $p$-test Exercise

I intended to answer Maddle’s $p$-test question, but T. Bongers has beaten me by two minutes, so I posted my answer here to save my work. The problem statement This is the sum: $$\sum\limits_{n=3}^\infty\frac{1}{n\cdot\ln(n)\cdot\ln(\ln(n))^p}$$ How do I tell which values of $p$ allow this to converge? The ratio test isn’t working out for me at all. Unpublished solution The integral test will do. $$ \begin{aligned} & \int_3^{+\infty} \frac{1}{x\cdot\ln(x)\cdot\ln(\ln(x))^p} \,dx \\ &= \int_3^{+\infty} \frac{1}{\ln(x)\cdot\ln(\ln(x))^p} \,d(\ln x) \\ &= \int_3^{+\infty} \frac{1}{\ln(\ln(x))^p} \,d(\ln(\ln(x))) \\ &= \begin{cases} [\ln(\ln(\ln(x)))]_3^{+\infty} & \text{if } p = 1 \\ \left[\dfrac{[\ln(\ln(x))}{p+1}]^{p+1} \right]_3^{+\infty} & \text{if } p \ne 1 \end{cases} \end{aligned} $$ When $p \ge 1$, the improper integral diverges. [Read More]