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I'm studying uniform convergence, and am looking for some examples of series $\sum f_n(x)$ that converge absolutely and uniformly in $[a,b]$, but $\sum |f_n(x)|$ does not converge uniformly in the same interval. So far I've found things like $\sum (1-x)x^{n}(-1)^n$ in $[0,1]$.

I don't want to start a big list; I'll accept whichever answer gives a couple of insightful examples.

Thanks!

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I assume you have a typo in your example, because otherwise $(1-x)$ is playing no role. –  Martin Argerami Feb 27 '12 at 0:40
    
@MartinArgerami The sum needs to converge at $x=1$. –  David Mitra Feb 27 '12 at 0:50
    
I totally missed that, good point. –  Martin Argerami Feb 27 '12 at 0:55

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