Primitive of a distribution

Question

I need some help with this exercise, about calculating the primitive of a distribution $T$ given by a series. Is the following:

$$<T,\phi>=\displaystyle\sum_{n=1}^{\infty}\phi(n)\;\;\;\;\phi\in\cal{D}(\mathbb{R})$$

Thanks a lot for any help.

Answer 1

To finally answer the question: Let $H_n(x)$ be the Heaviside function with the step 1 at $x=n$, i.e. $H_n(x)=0$ for $x<n$ and $H_n(x)=1$ for $x\geq n$. As $H_n'=\delta_n$, you have for a test function $\varphi$ $$ (\sum_n H_n)'(\varphi)=-(\sum_n H_n)(\varphi') = (\sum_n H_n')(\varphi) = \sum_n \delta_n(\varphi) = \sum_n \varphi(n) $$ as the considered sum is in fact finite due to the compact support of $\varphi$. So the general solution is $$ \sum_n H_n + \text{const}. $$