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If I wish to show that $\sum f_n (x)$ does not converge pointwise to a function, is it enough to show that for a particular $x$ in the domain, the sum is is divergent? Thanks. But if that were true then we can't have anything converging to say $1/x$? In particular, I am wondering if $\sum {1 \over n} sin (nx)$ does not converge to a function since it is a divergent sum for $x=0$.

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You haven't specified your mode of convergence. – kahen Sep 19 '11 at 16:10
What do toy consider the domain of $1/x?$ – jspecter Sep 19 '11 at 16:28
@kahen: Edited. – R B Sep 19 '11 at 17:11
@jspecter: I am not sure I understand the question... – R B Sep 19 '11 at 17:13
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Sorry. What do you consider the domain in $1/x?$ If it's $\mathbb{R} \setminus 0$ then you can surely have a series of functions converging to it, but those functions would be from $\mathbb{R} \setminus 0$ to $\mathbb{R}.$ – jspecter Sep 19 '11 at 17:22
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