Generalized Binomial Series

A useful application dealing witii power series e q)ansion regards the calculation of special limits, otherwise difficult to solve. As an example one can immediately evaluate  

The (Newton) binomial series expansion generally looks like 

Worth specializing this formula for the case when 

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How does the general form of this expression compare with the Debye-Hiickel limiting law as far as agreement with experiment is concerned To see what the extra term (1 -i- Bal does to the shape of the log/. versus I curve, one can expand it in the form of a binomial series... 

Generally, to calculate the probability of a certain number of successes in a series of trials, where the probability of success is constant, and where the trials are independent, it is appropriate to use the binomial probability mass function ... 

Clearly, any polynomial in the denominator can be expanded into an ascending series by synthetic division in the present case, we can use the binomial theorem written generally as... 

In Example 1.7 we defined composite events as pairs of elementary events. More generally, a composite event is a sequence of AT repetitions of independent elementary events. The probability of a specific sequence of n s and N n J s is given by Equation (1.7) (page 4). What is the probability that a series of N trials has n s and N - n J s in any order Equation (1.19) (page 12) gives the total number of sequences that have n s and N - n J s. The product of Equations (1.7) and (1.19) gives the probability of n 8 and N - n s irrespective of their sequence. This is the binomial distribution ... 

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