Let $X$ be a random variable taking values in $\mathbb{N}$ such that $\mathbb{P}(X = 0) > 0$, and let $H_{X}$ be its auxiliary power series:
$$H_{X}(t) = \ln(\mathbb{P}(X = 0)) + \sum_{k=1}^{\infty} \lambda_{k} t^{k}$$
We say that $X$ is $\lambda$-positive if $\lambda_{k} \geqslant 0$ for all $k \geqslant 1$. We assume that $X$ is $\lambda$-positive.
Show that, for all $t \in [-1,1], G_{X}(t) = \exp\left(H_{X}(t)\right)$ and that $\sum_{k=1}^{\infty} \lambda_{k} = -\ln(\mathbb{P}(X = 0))$.