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This may be an elementary question but I haven't found any confirmation of the following question or come up with any counterexample.

Suppose we have an open cover $\{U_{\alpha}\}$ for a manifold $M$ and smooth functions $f_{\alpha}:U_{\alpha}\to\mathbb{R}$. Does there exist a partition of unity subordinate to $\{U_{\alpha}\}$, say $\{\varphi_{\alpha}\}$, such that $$\sum_{\alpha}\varphi_{\alpha}f_{\alpha}$$ is smooth?

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    Let $\varphi_\alpha$ be any smooth partition of unity on $\{U_\alpha\}$. $\varphi_{\alpha}f_\alpha$ is a smooth function on $U_\alpha$, since $\varphi_\alpha$ has support lying in some compact subset of $U_\alpha$, you can identfiy $\varphi_\alpha f_\alpha$ with a function on all of $M$ by extending by $0$. Inside every $U_\alpha$ only finitely many $\varphi_\alpha$ are non-zero, so $\sum_\alpha \varphi_\alpha f_\alpha$ is locally always a finite sum of smooth functions. Since smoothness is a local property the sum is smooth on all of $M$. – 2017-01-31
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    @s.harp Your argument seems to make sense. Thanks. – 2017-02-01

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