## Test For Divergence: An Example

This post is in response to a user-submitted question. I’m going to keep it short and sweet since I just had eye surgery and am still half-blind, so computer tasks are somewhat difficult. The question gives the following infinite sum:  \begin{equation*} \sum_{n=1}^{\infty} \left(1+\frac{3}{n}\right)^{-5n} … Continue reading

## Trig Substitution and Completing the Square: An Example

This post is about another integration problem. It was submitted to the site by the same person who submitted this question. Part me wonders if this individual isn’t just getting to do their homework for them… But I will let it slide since I was … Continue reading

## Integration by Parts: An Example

This post is about a problem a student recently submitted to the website. I decided to answer it in post format, because to do so through email would look too messy. The question is the following: \begin{equation*} \int e^{2x}\sin(3x)dx \end{equation*} This is a challenging … Continue reading

## Gabriel’s Horn and Improper Integrals

This post is about an interesting problem in mathematics called Gabriel’s Horn (also known as Toricelli’s trumpet). It comes up when learning about surface area and improper integrals in calculus. The name “Gabriel’s Horn” is explained by the following quote, courtesy of Wikipedia: The name … Continue reading