Alternating Series

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alternating series

[′ȯl·tər·nād·iŋ ′sir·ēz]
Any series of real numbers in which consecutive terms have opposite signs.

Alternating Series


an infinite series whose terms are alternately positive and negative:

u1 - u2 + u3 - u4 + … + (-I)n-1un + …

for uk > 0. If the terms of an alternating series monotonically decrease (un+1 < un) and tend toward zero (lim un = 0), then the series is convergent (the Leibniz theorem). The remainder of the convergent series

rn = (-1)nun+1 + …

has the sign of its first term and is less than this term in absolute value. Some very simple examples of convergent alternating series are

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