there must be a close connection between (1) and the classical binomial series. The aim of this short article is to offer another proof of (1) and an extension of binomial series by means of telescoping method.
What is remarkable is that this identity generalizes, indeed, the well known binomial series because when c [right arrow] [infinity], it becomes
Among specific topics are Archimedes and the geometric series to be, the binomial series
in the hands of Euler, the Cauchy product, welcome to irrationals: the complete space of real numbers, and features of Ces[sz]ro and Abel means.
As [beta] [right arrow] 0, expanding the square root in Equation (6) in its binomial series
, we have [V.sub.1] [right arrow] 1/2 and then [Q.sub.1] [right arrow] 0.
The sampling data was fitted to a Poisson and to a negative binomial series
, respectively, by the dispersion index (I) and the dispersion parameter k, following Elliott (1983).
The product of the annual year coefficients from the two delta-lognormal model components, which individually indicated less decline than the unstandardized indices, provided final indices of abundance that declined 64% over the time series (due to the product of two positive fractional values <1 providing a even smaller value <1)--nearly identical to the unstandardized, Poisson, and negative binomial series
They cover initial encounters with combinatorial reasoning; selections, arrangements, and distributions; binomial series
and generating functions; alternating sums, the inclusion-exclusion principle, rook polynomials, and Fibonacci Nim; recurrence relations; special numbers; linear spaces and recurrence sequences; and counting with symmetries.