Find $f(x)$ to which the given sequence of functions converges

Question

$$f_{n}(x) = \begin{cases} \sin^{2}\pi x, & n≤ |x|≤n+1, \\ 0,& |x| < n \text{ or }|x|≥ n+1.\end{cases}$$

How can I find $f(x)$ to which $f_{n}(x)$ converges? I do always have problems with finding $f(x)$ for the sequence of functions given in the form of $n$. Is there any particular way to find $f(x)$ for such functions$?$

Answer 1

For each $x\in\Bbb R$, $f_n(x)=0$ for each $n\in\Bbb N$ with, at most, one exception. Therefore, $\lim_{n\to\infty}f_n(x)=0$.

And the convergence is not uniform, since you have$$(\forall n\in\Bbb N):f_n\left(n+\frac12\right)=1.$$