Let $s^{(\alpha)}=s_1^{\alpha_1}\dots s^{\alpha_n}_1$, $(\alpha_i\geq0,\ i=1,\dots,n)$ be the differentiation operator in the $n$-dimensional Mikusinski operational calculus and let $a_(k)$ be complex numbers depending on multi orders $(k)\in N^n_0$. The necessary and sufficient conditions for the normal convergence of power series \[ S=um_{(k)}a_{(k)}s^{(lpha k)},\quad((lpha k)=(lpha_1k_1,\dots,lpha_n,k_n)). \] in the space of $n$-dimensional Mikusinski operators are given. It is shown that the convergence depends on the quasi-analyticity of certain Lelong-Carleman class, which contains the factor of convergence. This completes the results of T. Boehme, J. Wloka, B. Stankovic and the author [1, 6, 9].