I am following stein's real analysis book and he claims that if $a_n=\frac{1}{2\pi}\int_{-\pi}^{\pi}f(x)e^{-inx}dx$ where $f\in L^1([-\pi,\pi])$ then $\sum_{n=-\infty}^{\infty} a_n r^{|n|}e^{inx}$ converges absolutely and uniformly for each $r$, $0\leq r <1$.

I know for a fact that $a_n$ is bounded because $f$ is integrable but am I missing out on a convergence test which validates the above claim? It seems like Abel's test and the Weierstrass M test don't work here...

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You use Weierstrass: if $0\leq r<1$, then $\left|a_nr^{|n|}e^{inx}\right|\leq Mr^{|n|},$ and the sum of $r^{|n|}$ converges.

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