I am learning about uniform convergence of function series. I wanted to ask whether I answered this exercise correctly:
Let $f_n=\frac{\sqrt{n}\cdot \sin(x)}{nx^{2}+2}$
- Does $f_n$ uniformly converge in $[0,\infty)$
- Does $f_n$ uniformly converge in $[-\pi,\pi]$?
- Does $\displaystyle\int_{\pi}^{2\pi}f_n(x)dx \rightarrow 0$?
Solution
$f_n$ is point-convergent to $f(x)=0$ in R. Put $x_n=\frac{1}{\sqrt{n}}$, then $|f_n(x_n)-f(x_n)|\rightarrow \frac{1}{3}$, and therefore the answer to 1. and 2. is that $f_n$ is not uniformly convergent in those intervals.
For 3., the answer is yes since $f_n(x)\rightarrow f(x)=0$.
What is odd about this solution to me is that you can answer 1. and 2. seemingly in the same way. So, is this correct?
Thanks!