# Thread: uniform convergence problem

1. ## uniform convergence problem

Let $S_n(x)= \displaystyle \frac{1}{n}e^{-n^2x^2}$. Show that there is a function, $S(x)$, such that $S_n(x)-->S(x)$ uniformly on R and that $S'_n(x)-->S'(x)$ for every x but that the convergence of the derivatives is not uniform in any interval which contains the orgin.

2. Originally Posted by wopashui
Let $S_n(x)= \displaystyle \frac{1}{n}e^{-n^2x^2}$. Show that there is a function, $S(x)$, such that $S_n(x)-->S(x)$ uniformly on R and that $S'_n(x)-->S'(x)$ for every x but that the convergence of the derivatives is not uniform in any interval which contains the orgin.
What have you done so far? The first step is obviously to find the limit function $S(x) = {\displaystyle\lim_{n\to\infty}}\,\frac1ne^{-n^2x^2}.$ So, what is that limit?