So the problem is:
Let $a_n,b_n\in\mathbb{R}$ and define $$f_n(x) = \frac{\sin(b_nx)}{1+(n^3x-a_n)^2}.$$ Prove that there exists $f\in L^1(\mathbb{R})$ such that $f=\sum_{n=1}^{\infty}f_n$ almost everywhere and in norm.
I think the best way to do this would be using the monotone convergence theorem, but I am not sure how to integrate the $f_n$/show that their integrals are bounded. Also, the way we defined $f\in L^1(\mathbb{R}^N)$ is that there exists a sequence of simple functions $f_n$ such that
$$\sum_{n=1}^{\infty} \int |f_n|<\infty;$$ $$f(x) = \sum_{n=1}^{\infty} f_n(x) \text{ for every } x\in\mathbb{R}^N \text{ such that } \sum_{n=1}^{\infty}|f_n(x)|<\infty.$$ Then $f\in L^1$.
I don't have much work done so any hints would be much appreciated; thank you!
$\int |f_n(x)| \leq \int \frac 1 {1+(n^{3}x-a_n)^{2}} \, dx =\frac 1 {n^{3}}\int \frac 1 {1+y^{2}} \, dy =\frac {\pi} {n^{3}}$ where we have used the substitution $y=n^{3}x-a_n$. Hence $\sum \int |f_n(x)| <\infty$. This implies $f=\sum \int f_n(x)$ exists a.e. and $f \in L^{1}$.