Alternating Series (redirected from Leibniz' estimate for alternating series)

alternating series [′ȯl·tər·nād·iŋ ′sir·ēz]

(mathematics)

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

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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