= Solution
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A <series> $\sum_{n=1}^\infty a_n$ converges when its <partial sums> $s_N=\sum_{n=1}^N a_n$ have a finite <limit of a sequence>.
The <comparison test> says that if $0\leq a_n\leq b_n$ eventually and $\sum b_n$ converges, then $\sum a_n$ converges. Indeed, the <partial sums> of the nonnegative terms $a_n$ are increasing, and are bounded above by a fixed initial contribution plus the sum of the comparison <series>. The <bounded monotone sequence theorem> proves their convergence. Finite initial terms do not affect convergence. Applying this result to $|a_n|$ also proves <absolute convergence> whenever $|a_n|\leq b_n$ eventually with $\sum b_n<\infty$.
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