following formula is sometimes called Binomial 3. It is the difference
between two different numbers raised to the same power
Examples are: [1.2]
Assume [2.1] (Method of Undetermiate Coefficients)
The a's are numbers independent of x and y.
Multiply both sides by (x-y): [2.2]
Note: , and is just a convenient way of writing he terms. For instance the coefficient of y is a2x-a1
Equating coefficients, we find: [2.3]
Substituting these values in 2.1, we find: [2.4]
In 2.4, when n is a positive integer, and the value of x approache that of y, then: [3.1] Noting that the limit is nxn-1.
Let n=p/q, where p and q are nonnegative integers. Then [3.2]
Rearranging 3.2: [3.3]
Writing x1/q=x and y1/q=y. Also dividing top and bottom by (x-y): [3.4]
Taking the limit (recalling that p is a positive integer): [3.5]
Reverting to n=p/q and x=x1/q: [3.6]
So, once again, the limit is nxn-1.
Let n=p, where p is a positive integer, so: [3.7]
Multiplying by -1 to bring the right-hand side into our normal form, with a positve xp: [3.9]
Taking the limit, and noting 1/x2p=x-2p: [3.10]
Re-substituting n=-p [3.11] So, once again, the limit is nxn-1. Therefore, we conclude that [3.12]
r is a rational number. We can make a weak claim, that if 3.12 is true
for all rational numbers, then it is most likely true for all real
numbers (which it actually is, although we have not proved it).