Limit of a distribution (Sum of Deltas)

Question

I have to find the limit of $\langle T_n\mid \varphi\rangle$ in $D^*$ when n goes infinity, with $T_n=\frac {1}{n} \sum_{k=0}^{n} \delta_\frac{k}{n}$

Any idea?

Answer 1

if $\varphi$ is a smooth function, then $<T_n|\varphi>$ is just an $(n+1)/n$ times the $n$ point numerical integral approximation of $$ \int \limits_{0}^{1} \varphi(x) dx $$ so that's what it will converge to.