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working over the reals, does uniform convergence imply absolute convergent? I'm having troubles proving this or finding a counter example. I have found a counter-example when working over a closed interval [a,b] but I cannot find one for $\mathbb{R}$. What I'm asking is, if I had a sequence of functions $f_n$ over the reals such that $\sum_n f_n$ converges uniformly does $\sum_n |f_n|$ converge absolutely for any $x \in \mathbb{R}$

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Why not just define $f_n = (-1)^n/n$ to be a constant function, to get a counter-example.

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  • $\begingroup$ Can we do that? I don't see how that is uniformly convergent though. Also, it doesn't depend on $x$? $\endgroup$ – jinggula89 Mar 7 '15 at 21:30
  • $\begingroup$ You can't get more uniform behavior than with constant functions. They behave in every point of the domain in exactly the same way. $\endgroup$ – Lutz Lehmann Mar 8 '15 at 16:32

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