If eventually and converges, then converges. The nondecreasing partial sums of the nonnegative are bounded above by a finite initial contribution plus the sum of ; bounded monotone sequence theorem proves the claim. Consequently with summable proves absolute convergence of . Finite changes to initial terms do not affect convergence.
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The Direct Comparison Test is a method used in calculus to determine the convergence or divergence of an infinite series. It compares the series in question with a known benchmark series whose convergence behavior is already established. This test is particularly useful when dealing with series that have positive terms.