  Let $A$ be a non-projectively-pluripolar set in a Fr\'{e}chet space $E.$ We
give sufficient conditions to ensure the convergence on some zero-neighbourhood
in $E$ of a (sequence of) formal power series of Fr\'{e}chet-valued continuous
homogeneous polynomials provided that the convergence holds at a
zero-neighbourhood of each complex line $\ell_a := \C a$ for every $a \in A.$
