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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


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

© 1996-9 Eric W. Weisstein