I'm reading Fourier analysis an introduction by Stein, and I have a problem from section 5.4 about the Poisson kernel. For the following equations
\begin{align} A_{r}(f)(\theta)&=\sum_{n=-\infty}^{\infty}r^{|n|}a_{n}e^{in\theta} \\ &=\sum_{n=-\infty}^{\infty}r^{|n|} \bigg(\frac{1}{2\pi}\int_{\pi}^{\pi}f(\varphi)e^{-in\varphi}d\varphi \bigg)e^{in\theta} \\ &=\frac{1}{2\pi}\int_{-\pi}^{\pi}f(\varphi) \bigg(\sum_{n=-\infty}^{\infty}r^{|n|}e^{in(\varphi-\theta)} \bigg)d\varphi \end{align}
I don't understand why we can interchange the integral and infinite sum in the last equation. The text says it is "justified by the uniform convergence of the series." I am not sure which series it means and why uniform convergence can justify this interchange.