Removing terms from the expansion of an analytic function of several variables is still analytic?

Question

Suppose $f(s,t) = \sum_{p,q} a_{p,q} s^p t^q$ is analytic in a neighborhood of $(s,t)=(0,0)$. If we let $b_{p,q}$ either equals $0$ or $a_{p,q}$, does there exist such $b_{p,q}$ so that $g(s,t) = \sum_{p,q} b_{p,q} s^p t^q$ is no longer analytic in any neighborhood of $(s,t)=(0,0)$?