Assume we have a series of functions which converge uniformly on a closed interval. Does that implies absolutely convergence (pointwise)?
2026-02-26 20:04:37.1772136277
Does uniform convergence of a series implies pointwise *absolutely* convergence?
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No. Take any real sequence $a_n$ s.t. $\sum_n a_n$ converges, but does not converge absolutely (e.g. $a_n=(-1)^n 1/n$). Then consider $f_n(x) := a_n$. Obviously, the convergence is uniform in $x$, because nothing depends on $x$.