A function series $\sum f_n(x)$ is pointwise convergent in $A_p$ if $\forall x\in A_p$, $\sum f_n(x)$ converges.

It is totally convergent in $A_p$ if it passes the Weierstrass M-test. If a series passes the M-test, then it is also uniformly and pointwise convergent.

I have problems when I have to determine the uniform convergence when a series is not totally convergent.

In practice, what methods can I use to test the uniform convergence without Weierstrass M-test?


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