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A nonnegative series is controlled by a larger or smaller nonnegative series.
Comparison test: Let and be series with and for all .
This test reduces many convergence questions to comparisons with standard benchmark series, and it is complemented by the limit comparison test when direct inequalities are hard to establish.
A series is a formal expression together with the associated partial sums .
Whether a series has a value is determined by the behavior of its partial sums: it either converges or diverges. Many standard tools for deciding this are collected as convergence tests such as the ratio test and the comparison test.
A convergent series is a series whose partial sums converge to a finite limit , in which case one writes .
A divergent series is a series whose partial sums do not converge to a finite limit, i.e. it is not a convergent series.
Divergence may happen because partial sums grow without bound or because they oscillate. A basic necessary condition for convergence is terms go to zero, whose contrapositive often certifies divergence quickly.