Alternating Series

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alternating series

[′ȯl·tər·nād·iŋ ′sir·ēz]

(mathematics)

Any series of real numbers in which consecutive terms have opposite signs.

The following article is from The Great Soviet Encyclopedia (1979). It might be outdated or ideologically biased.

Alternating Series

an infinite series whose terms are alternately positive and negative:

u₁ - u₂ + u₃ - u₄ + … + (-I)^n-1u_n + …

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

r_n = (-1)ⁿu_n+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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