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Are there any intuitive tests that might help one decide whether a sequence of functions converges / converges uniformly? For example, an intuitive test I have recently realized for uniform continuity is that the graph of the function must not tend to having infinite slopes, and that helped. Any suggestions will be very much appreciated. Thanks.

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For sequences, I'm afraid there is none. For series, you have Weierstrass M-test: if a series of functions $\sum f_n(x)$ is such that $\lvert f_n(x) \rvert \le M_n$ for a summable numerical sequence (i.e. $\sum M_n < \infty$), then $f_n(x)$ converges uniformly. –  Giuseppe Negro Sep 18 '11 at 8:25
    
Thanks, Giuseppe. –  I. S. Sep 18 '11 at 20:23

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