Summation of series
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Let us begin by restricting attention to nonnegative series (i.e., series whose terms are all nonnegative). We remind the reader that convergence is a tail property of a series. So the investigation of the convergence of a series
a0 + a1 + a2 + a3 + · · · depends only on the properties of
am + am+1 + am+2 + am+3 + · · · for any positive integer m. This is particularly important for series where some test for convergence may be applied, but the conditions of the test only hold for terms that are suﬃciently far out in the tail.