Binomial coefficient
The coefficients for the terms in the expansion of the binomial theorem, with broken exponents, can be written as
The upper fraction m/n is the exponent of the binomial theorem, the lower number k is the running number of the term in the outcome.
Explanation
We are going to expand the table for the development of binomial coefficients with broken exponents. A number arises from the sum of the number just below it and to the left of it. You can see it in the penultimate row at
All rows have an infinite number of terms.
m/n 0 1 2 3 4 5 k → ∞ ··· ··· ··· ··· ··· ··· ··· ··· 5/2 ··· 3/2 ··· 1/2 ··· 0/2 ◦ ◦ ◦ ◦ ◦ ··· −1/2 ··· −3/2 ··· ··· ··· ··· ··· ··· ··· ··· ···
For broken exponents we have the formula
HistoryIn 1676, the English mathematician Isaac Newton gave the following information about his formula in a letter in which A, B, C, … always indicates the immediately preceding term. With this we are going to calculate the square root of and get Nowadays we work with Taylor series. The series for the square root gives of course exactly the same solution. |