I am trying to find a dominating function $g(x)$ for the series of functions $\{f_n\}$ $$f_n = \frac{x~e^{-x/n}}{n^2}$$ The integral of the function $f_n$ I believe is a point mass of $1$ at $0$ when $n \rightarrow \inf$, But the trouble is how to a find a dominating function which can always be greater then $f_n$ and is finitely integrable, for all $n$, since $f_n$ keeps growing in height and moving toward $0$.
Any help would be greatly appreciated.
Note this is a function that I have made up to help me understand the application of Lebesgue's Dominating convergence theorem.
Actually, your functions become flatter for $n \to \infty$. But since $\int_0^\infty f_n(x)\,dx = 1$ independent of $n$, and $f_n \to 0$ uniformly, there cannot be an integrable dominating function.
If there were an integrable dominating function, the integrals of $f_n$ would converge to the integral of the pointwise limit, which is $0$.
You would get a peak marching towards $0$ for $g_n(x) = n^2xe^{-nx}$, but again, since the integral is independent of $n$, and $g_n \to 0$ pointwise, there cannot be an integrable dominating function.