Weierstrass M-Test

Let $\sum_{k=1}^\infty u_n(x)$ be a Series of functions all defined for a set $E$ of values of $x$. If there is a Convergent series of constants

$$\sum_{n=1}^\infty M_n,$$

such that

$$\vert u_n(x)\vert \leq M_n$$

for all $x \in E$, then the series exhibits Absolute Convergence for each $x \in E$ as well as Uniform Convergence in $E$.

See also Absolute Convergence, Uniform Convergence

References

Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 301-303, 1985.