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power series

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Definition

A power series in a variable XX and with coefficients in a ring RR is a series of the form

n=0 a nX n \sum_{n = 0}^\infty a_n X^n

where a na_n is in RR for each n0n\ge 0. Given that there are no additional convergence conditions, a power series is also termed emphatically as a formal power series. If RR is commutative, then the collection of formal power series in a variable XX with coefficients in RR forms a commutative ring denoted by R[[X]]R [ [ X ] ].

More generally, a power series in kk commuting variables X 1,,X kX_1,\ldots, X_k with coefficients in a ring RR has the form n 1=0,n 2=0,,n k=0 a n 1n kX 1 n 1X 2 n 2X k n k\sum_{n_1=0,n_2=0,\ldots, n_k = 0}^\infty a_{n_1\ldots n_k} X_1^{n_1} X_2^{n_2}\cdots X_k^{n_k}. If RR is commutative, then the collection of formal power series in kk commuting variables X 1,,X kX_1,\ldots, X_k form a formal power series ring denoted by R[[X 1,,X k]]R [ [ X_1,\ldots, X_k ] ].

More generally, we can consider noncommutative (associative unital) ring RR and words in noncommutative variables X 1,,X kX_1,\ldots, X_k of the form

w=X i 1X i m w = X_{i_1}\cdots X_{i_m}

(where mm has nothing to do with kk) and with coefficient a wRa_w \in R (here ww is a word of any length, not a multiindex in the previous sense). Thus the power sum is of the form wa wX w\sum_w a_w X_w and they form a formal power series ring in variables X 1,,X kX_1,\ldots, X_k denoted by RX 1,,X kR\langle \langle X_1,\ldots, X_k \rangle\rangle. Furthermore, RR can be even a noncommutative semiring in which case the words belong to the free monoid on the set S={X 1,,X k}S = \{ X_1,\ldots, X_k\}, the partial sums are then belong to a monoid semiring RSR\langle S\rangle. The formal power series then also form a semiring, by the multiplication rule

ra rX rb sX s= w u,v;w=uva ub vX w \sum_{r} a_r X_r \cdot \sum b_s X_s = \sum_w \sum_{u,v; w = u v} a_u b_v X_w

Of course, this implies that in a specialization, bb-s commute with variables X i kX_{i_k}; what is usually generalized to take some endomorphisms into an account (like at noncommutative polynomial level of partial sums where we get skew-polynomial rings, i.e. iterated Ore extensions).

Examples

Polynomials

For a natural number kk, a power series n=0 a nX n\sum_{n=0}^\infty a_n X^n such that a n=0a_n = 0 for all n>kn \gt k is a polynomial of degree at most kk.

Taylor series

MacLaurin series

For fC ()f \in C^\infty(\mathbb{R}) a smooth function on the real line, and for f (n)C ()f^{(n)} \in C^\infty(\mathbb{R}) denoting its nnth derivative its MacLaurin series (its Taylor series at 00) is the power series

n=0 1n!f (n)(0)x n. \sum_{n = 0}^\infty \frac{1}{n!} f^{(n)}(0) x^n \,.

If this power series converges to ff, then we say that ff is analytic.

Laurent series

Puiseux series

References

A formalization in homotopy type theory and there in Coq is discussed in section 4 of

Revised on March 13, 2014 05:37:59 by Urs Schreiber (88.128.80.11)