In such cases the algebra of formal ability series is the full algebra of the monoid of purely natural numbers over the underlying expression ring.[fifteen] When the fundamental term ring is usually a differential algebra, then the algebra of formal energy series is also a differential algebra, with differentiation done term-by-phrase. Laurent series[edit]
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Chemistry. a group of connected chemical features arranged so as of raising atomic quantity: the lanthanide series.
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An arithmetico-geometric series can be a generalization in the geometric series, that has coefficients on the common ratio equal on the conditions in an arithmetic sequence. Case in point:
The notion of series is often conveniently extended to the case of the seminormed Place. If xn is often a sequence of aspects of a normed House X and if x is in X, then the series ∑xn converges here to x in X In the event the sequence of partial sums in the series ( ∑ n = 0 N x n ) N = one ∞ textstyle left(sum _ n=0 ^ N x_ n ideal)_ N=1 ^ infty
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is usually approximated with rational numbers (and that is unknown as of nevertheless). Much more specially, the values of n with large numerical contributions on the sum would be the numerators on the continued fraction convergents of π displaystyle pi
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