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Weierstrass M-test (Theorem)

Let $ X$ be any set, $ \{f_n\}_{n\in\mathbb{N}}$ a sequence of real or complex valued functions on $ X$ and $ \{M_n\}_{n\in\mathbb{N}}$ a sequence of non-negative real numbers. Suppose that, for each $ n \in \mathbb{N}$ and $ x \in X$, we have $ \vert f_n(x)\vert \le M_n$. Then $ f=\sum_{n=1}^{\infty} f_n$ converges uniformly if $ \sum_{n=1}^{\infty} M_n$ converges.



"Weierstrass M-test" is owned by vypertd. [ full author list (2) | owner history (1) ]
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See Also: convergent series


Attachments:
proof of Weierstrass M-test (Proof) by CWoo
Weierstrass M-test for continuous functions (Corollary) by CWoo
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Cross-references: converges, converges uniformly, functions, complex, real, sequence
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This is version 10 of Weierstrass M-test, born on 2002-08-14, modified 2006-08-11.
Object id is 3293, canonical name is WeierstrassMTest.
Accessed 4355 times total.

Classification:
AMS MSC: 30A99 (Functions of a complex variable :: General properties :: Miscellaneous)
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