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  2. Formal power series - Wikipedia

    en.wikipedia.org/wiki/Formal_power_series

    A formal power series can be loosely thought of as an object that is like a polynomial, but with infinitely many terms.Alternatively, for those familiar with power series (or Taylor series), one may think of a formal power series as a power series in which we ignore questions of convergence by not assuming that the variable X denotes any numerical value (not even an unknown value).

  3. Generating function - Wikipedia

    en.wikipedia.org/wiki/Generating_function

    Unlike an ordinary series, the formal power series is not required to converge: in fact, the generating function is not actually regarded as a function, and the "variable" remains an indeterminate. One can generalize to formal power series in more than one indeterminate, to encode information about infinite multi-dimensional arrays of numbers.

  4. Power series - Wikipedia

    en.wikipedia.org/wiki/Power_series

    The partial sums of a power series are polynomials, the partial sums of the Taylor series of an analytic function are a sequence of converging polynomial approximations to the function at the center, and a converging power series can be seen as a kind of generalized polynomial with infinitely many terms. Conversely, every polynomial is a power ...

  5. Probability-generating function - Wikipedia

    en.wikipedia.org/wiki/Probability-generating...

    The probability generating function is an example of a generating function of a sequence: see also formal power series. It is equivalent to, and sometimes called, the z-transform of the probability mass function.

  6. Formal calculation - Wikipedia

    en.wikipedia.org/wiki/Formal_calculation

    In a formal power series, the powers of the variable are used only as position-holders for the coefficients, so that the coefficient of is the fifth term in the sequence. In combinatorics, the method of generating functions uses formal power series to represent numerical sequences and multisets, for instance allowing concise expressions for ...

  7. Function series - Wikipedia

    en.wikipedia.org/wiki/Function_series

    Each type of convergence corresponds to a different metric for the space of functions that are added together in the series, and thus a different type of limit. The Weierstrass M-test is a useful result in studying convergence of function series.

  8. Indeterminate (variable) - Wikipedia

    en.wikipedia.org/wiki/Indeterminate_(variable)

    A formal power series in an indeterminate is an expression of the form + + + …, where no value is assigned to the symbol . [7] This is similar to the definition of a polynomial, except that an infinite number of the coefficients may be nonzero.

  9. Series (mathematics) - Wikipedia

    en.wikipedia.org/wiki/Series_(mathematics)

    While many uses of power series refer to their sums, it is also possible to treat power series as formal sums, meaning that no addition operations are actually performed, and the symbol "+" is an abstract symbol of conjunction which is not necessarily interpreted as corresponding to addition. In this setting, the sequence of coefficients itself ...