# 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:

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

References in periodicals archive ?
Then we use the expression for the power sum polynomial as an alternating sum of hook Schur polynomials.
We next rewrite the power sum symmetric function as an alternating sum of hook Schur functions
We push forward the resulting alternating sum to [[LAMBDA].
We define our quantum power sum polynomial in analogy with the classical case as the alternating sum of quantum hook Schur polynomials,
The main ingredient is Rice's formula Flajolet and Sedgewick (1995) which allows to write an alternating sum as a contour integral:
For the asymptotic evaluation, we use a contour integral representation of alternating sums ("Rice's method").
Proof of Theorem 2: The exactness of the sequence (1) gives us that the alternating sum of dimensions is 0.
The terms [alpha] and [beta] are typical expressions to which 'Rice's method' can be applied, due to the presence of the binomial coefficient inside an alternating sum.
Now we look at the alternating sum labelled (b) which has a double pole at z = 1 and a simple pole at z = 2.
Now Rice's method can again be used to approximate the alternating sums.

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