= Direct comparison test
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= Comparison test
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If $0\le a_n\le b_n$ eventually and $\sum b_n$ converges, then $\sum a_n$ converges. The nondecreasing partial sums of the nonnegative $a_n$ are bounded above by a finite initial contribution plus the sum of $b_n$; <bounded monotone sequence theorem> proves the claim. Consequently $|c_n|\le b_n$ with summable $b_n$ proves <absolute convergence> of $\sum c_n$. Finite changes to initial terms do not affect convergence.
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