Convergence in $L^p(\Bbb R)$

Question

I still want to examine different convergence results of the sequence of functions $f_n(x)= e^{-\frac{x^2}{n^2}}$ on $\mathbb{R}$. Convergence in the sense of distributions is shown in the following post: Convergence in $\mathcal{D}'(\mathbb{R})$ . Now I want to check the convergence in Convergence in $L^p(\mathbb{R})$ for $1\leq p \leq \infty$. How would you start the analysis? Guess for a limit (in this case $1$) and then compute $\vert\vert f_n -1\vert \vert_{L^p(\mathbb{R})}$? Or is there another trick or theorem with an equivalent condition...?

Answer 1

There is a problem. The constant function $1$ is not in $L^p(\Bbb R)$ if $p\in[1,\infty)$, and $\|f_n-1\|_p=\infty$ for all $n$.

If $p=\infty$, $f_n$ converges pointwise to $1$, but not in the $L^\infty$ norm, since$\|f_n-1\|_\infty=1$ for all $n$.