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A convergence test for alternating series with decreasing term magnitudes tending to zero.
Alternating series test (Leibniz): Let be a sequence of real numbers such that
A common consequence is an error bound for partial sums: if is the limit and , then . This test is a basic source of conditional convergence and uses the necessary condition terms go to zero.
A series is a formal expression together with the associated partial sums .
A convergent series is a series whose partial sums converge to a finite limit , in which case one writes .
For a sequence of real or complex numbers, the th partial sum is the finite sum , for . Equivalently, and .
A conditionally convergent series is a series that is convergent but not absolutely convergent, meaning that converges while diverges.
Terms go to zero: If the series converges, then
Equivalently (the divergence test), if does not tend to , then the series is divergent. This is a basic prerequisite for many convergence tests and is often used to quickly rule out convergence before applying finer criteria like the root test or ratio test.