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Probability generating functions are often employed for their succinct description of the sequence of probabilities Pr(X = i) in the probability mass function for a random variable X, and to make available the well-developed theory of power series with non-negative coefficients.
A pancake number is the minimum number of flips required for a given number of pancakes. In this form, the problem was first discussed by American geometer Jacob E. Goodman. [1] A variant of the problem is concerned with burnt pancakes, where each pancake has a burnt side and all pancakes must, in addition, end up with the burnt side on bottom.
An infinite series of any rational function of can be reduced to a finite series of polygamma functions, by use of partial fraction decomposition, [8] as explained here. This fact can also be applied to finite series of rational functions, allowing the result to be computed in constant time even when the series contains a large number of terms.
Alternatively, the equality can be justified by multiplying the power series on the left by 1 − x, and checking that the result is the constant power series 1 (in other words, that all coefficients except the one of x 0 are equal to 0). Moreover, there can be no other power series with this property.
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 ...
The sum of powers of two from zero up to a given positive integer power is 1 less than the next power of two (i.e. the next Mersenne number). The simple, brute-force solution is just to manually double and add each step of the series:
Two cases arise: The first case is theoretical: when you know all the coefficients then you take certain limits and find the precise radius of convergence.; The second case is practical: when you construct a power series solution of a difficult problem you typically will only know a finite number of terms in a power series, anywhere from a couple of terms to a hundred terms.
The pancake number, which is the minimum number of flips required to sort any stack of n pancakes has been shown to lie between 15 / 14 n and 18 / 11 n (approximately 1.07n and 1.64n,) but the exact value remains an open problem. [10] In 1979, Bill Gates and Christos Papadimitriou [11] gave an upper bound of 5 / 3 n.