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Suppose that $\sum_{k=1}^\infty g_k(x)$ is a series of continuous functions $g_k$ on $[a,b]$ that converges uniformly to $g$ on $[a,b]$. Prove that \[ \int_a^b g(x) dx = \sum_{k=1}^\infty \int_a^b g_k(x) \ dx. \]
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Mon, 31 Mar 2014 03:58 GMT