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## Homework Statement

I want to justify that [itex] \int_{0}^{1} \frac{f(x)}{1-x} \ dx = \int_{0}^{1} f(x) \sum_{k=0}^{\infty} x^{n} \ dx = \sum_{k=0}^{\infty} \int_{0}^{1} f(x) x^{n} \ dx [/itex]

## Homework Equations

## The Attempt at a Solution

I always thought changing the order of summation and integration could be justified by uniform convergence. But the geometric series [itex] \sum_{k=0}^{\infty} x^{n} [/itex] does not converge uniformly for [itex] x= 1 [/itex]. In fact, it doesn't converge at all for [itex] x=1[/itex]. Is that an issue?