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An infinite sum understood through its sequence of partial sums.
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 partial sum is a finite sum associated to a series .
The sequence encodes the series: statements about convergence or divergence of the series are statements about whether the partial sums form a sequence with a limit, as formalized in convergent series.
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.
Ratio test: Let be a series with for all sufficiently large , and define
Comparison test: Let and be series with and for all .