= Dirichlet test
{c}
If the partial sums of $\sum b_n$ are bounded and $a_n$ decreases to zero, then $\sum a_nb_n$ converges. The uniform version holds for functions $b_n(x)$ when their partial sums have one bound $M$ independent of $x$. Indeed partial sums starting at index $N$ have absolute value at most $2M$, and summation by parts bounds every weighted tail from $N$ to $K$ by $2Ma_N$. The <uniformly Cauchy sequence> criterion proves <uniform convergence>. This justifies integrating a conditionally convergent <Fourier series> on intervals staying away from its endpoint jump.
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