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With the Weierstrass M-test for uniform convergence where for a functions $f_r:A \to \mathbb{R}$ you find an $M_r \in \mathbb{R}$ where $\forall x \in A, |f_r(x)| \leq M_r $. Can the $M_r$ be also a function of $x$? Or do you have to essentially find the $\|f_r\|_\infty$ each time?

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up vote 1 down vote accepted

The constant has to be chosen independently of $x$, though it is sufficient to find any sequence of constants which bound the functions and converge in sum. Often you will indeed be finding the supremum norm, but sometimes there will be a less precise, more obvious bound that is more convenient. For example for $x\in[0,1]$ we can show that

$$ \sum_{n=1}^{\infty} \frac{\sin (x/n) \cos(4x/3n)}{n^2} $$ converges uniformly almost straight away, but the precise supremum norm is not so immediate.

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