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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.
For a sequence of real or complex numbers, the th partial sum is the finite sum , for . Equivalently, and .
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
In particular, if the ratio has a limit , the series converges absolutely for , diverges for , and the test is inconclusive for .
Comparison test: Let and be series with and for all .
For complex numbers , the geometric series with initial term and ratio is the series
Its partial sum through index is
For , subtracting times the finite sum from the sum proves this identity by cancellation.