Prove it is a distribution

Question

This question seriously bothering me, because there is one part I really don`t know how even to start thinking of it:

Define

1. $$f(x)=e^{1/x} \text{ for } x\in R-\{0\}$$

2. $$\lt \lambda,\phi\gt= \sum_{k=1}^{\infty} \phi^{(k)} \left(\frac{-1}{2}\right)^k$$

show that this defines $\lambda$, and $f(x)$ as a distribution on $R-\{0\}$ and prove that $\lambda$ is not of finite order. Show that there does not exist any distribution $g \in D'(R)$ such that $g=f$ on $R-\{0\}$

I`m totally confused with the part there is no $g$ distribution and do not know how to start solving it.

Thanks