I was doing a problem where I had to show that the integral is convergent to prove the existence of the expected value of a probability density function. But I don't fully understand the solution:
$ \frac{|x|e^x}{(e^x + 1)^2} \sim_{-\infty} |x|e^{-x} $ and $ \int_{-\infty}^{0} |x|e^x $ converges by comparing to $\frac{1}{x^2}$
How do we get the idea of using $\frac{1}{x^2}$ ? Is it just being used to solve these kinds of problems or is there a specific technique ?
Because for $x<0$ we have $xe^x<\frac{1}{x^2}$ and $1/x^2$ converges on $(-\infty,-\varepsilon]$ for positive $\varepsilon$.