Let the sequence of fuctions $f_n$ such that $f_n:[0,1] \rightarrow R$ where $f_n(x)=nxe^{-nx^2}$. Show that the sequence is convergent pointwise to $f$.
Futhermore show that $f$ is R-Integrable and $\int_0^1 f =0$. Show that
$$
\lim_{n \rightarrow +\infty}\int_0^1f_n=\frac{1}{2}
$$
Finally, show that the convergence is not uniform.
MY ATTEMPT
If $x \in ]0,1]$
$$
\lim_{n\rightarrow +\infty}{nxe^{-nx^2}}=x\lim_{n\rightarrow +\infty}{\frac{n}{e^{nx^2}}}=\frac{1}{x}\lim_{n\rightarrow+\infty}{\frac{1}{e^{nx^2}}}=0
$$
If $x =0$, so $\lim_{n\rightarrow+\infty}{nxe^{-nx^2}}=0$
Therefore, the sequence is convergent pointwise to $f=0$. So $\int_0^1f=0$, $f$ is R-integrable (is a step fuction) and
$$
\lim_{n \rightarrow +\infty}\int_0^1 f_n= \frac{1}{2}
$$
solving by substitution.
MY ANSWER: How to show that the convergence is not uniform?
In the next exercise, was given the function $$f_n(x)= \frac{n^2x}{1+n^3x^2}$$
He asks to show the convergence is pointwise and show that the convergence is not uniform, the strategy is the same?
Best Answer
Hint: For uniformly convergent sequences of Riemann integrable functions, the Riemann integral behaves like $\lim_{n \to \infty} \int f_n = \int \lim_{n\to \infty} f_n $.