# When can a series be integrated term by term?

I have a function that is defined as a harmonic series, I would like to integrate it over part of its domain. I have been doing this by integrating term by term and summing the result, but I seem to remember something in little Rudin that gave conditions under which this is valid, however, if I could remember what it said I still don't think I understood it. So my question is when is the following true?

$$\int_a^b{\sum_{i}{f_i\left(x\right)}}=\sum_i{\int_a^b{f_i\left(x\right)}}$$

If the series is uniformly convergent and each $f_{n}(x)$ is integrable, then the formula works. I think there may be examples of pointwise- (but not uniformly-) convergent series for which the formula doesn't work, but I can't seem to find them at the moment.