I'm studyng a Book about $p-$adic numbers, and I have troubles with a "degenerate" case of a Newton polygon. Let $f(X)=\sum a_{i}X^{i}\in\mathbb{Q}_{p}[\![X]\!]$, we define the Newton poligon of $f$ as the convex hull of the set of points $(i,v_{p}(a_{i}))$ (where $v_{p}$ is the $p-$adic valuation) and if $a_{i}=0$ we introduce the point at infinity on the positive Y-axis $Y_{+\infty}=(a_{i},v_{p}(a_{i}))$. My trouble comes from this series: $$\sum_{n=0}^{\infty}p^{-n^{2}}X^{n}$$
I have a first infinite line, the positive part of the Y-axis, and since I have a strictly decreasing sequence on the $v_{p}(p^{n-^{2}})'s$ we cannot have a convex hull, so I only have the positive Y-axis? But the convex hull must cointain all of these points, so, my intuition suggest that the Newton poligon must be all the Y-axis
What is the newton polygon in this "degenerate" case? In the book there is not a discussion about this kind of examples, only treats the case of convergent power series, so we have a finite or infinite number of sides in our Newton polygon ...
Thanks a lot; I hope to have explained myself