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The Weierstrass M-test is a theorem stating that if an infinite series of functions $\sum f_n(z)$ is defined on a region $A$, and the absolute value of each function is bounded by a constant $M_n$ (i.e., $|f_n(z)| \le M_n$ for all $z \in A$), and if the numerical series $\sum M_n$ converges, then the original series $\sum f_n(z)$ converges uniformly on $A$. It provides a shortcut by reducing the problem of uniform convergence of a function series to the simpler problem of convergence of a numerical series of positive constants.

Conditions: The functions $f_n(z)$ are defined on a common region $A$.; There exists a sequence of non-negative real numbers $M_n$.; $|f_n(z)| \le M_n$ for all $n$ and all $z \in A$.; The numerical series $\sum_{n=1}^{\infty} M_n$ converges.

The Weierstrass M-test is a theorem stating that if an infinite series of functions $\sum f_n(z)$ is defined on a region $A$, and the absolute value of each function is bounded by a constant $M_n$ (i.e., $|f_n(z)| \le M_n$ for all $z \in A$), and if the numerical series $\sum M_n$ converges, then the original series $\sum f_n(z)$ converges uniformly on $A$. It provides a shortcut by reducing the problem of uniform convergence of a function series to the simpler problem of convergence of a numerical series of positive constants.

Conditions: The functions $f_n(z)$ are defined on a common region $A$.; There exists a sequence of non-negative real numbers $M_n$.; $|f_n(z)| \le M_n$ for all $n$ and all $z \in A$.; The numerical series $\sum_{n=1}^{\infty} M_n$ converges.