# 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.

## Alternating Series

an infinite series whose terms are alternately positive and negative:

*u _{1}* -

*u*+

_{2}*u*-

_{3}*u*+ … + (-I)

_{4}^{n-1}

*u*

_{n}+ …

*for u _{k}* > 0. If the terms of an alternating series monotonically decrease (

*u*<

_{n+1}*u*) and tend toward zero (lim

_{n}*u*= 0), then the series is convergent (the Leibniz theorem). The remainder of the convergent series

_{n}*r _{n}* = (-1)

^{n}u

_{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