The
following formula is sometimes called Binomial 3. It is the difference
between two different numbers raised to the same power
[1.1] Or

Examples are: [1.2]

[1.3] [1.4]

Proof

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 a_{2}x-a_{1}

Equating coefficients, we find: [2.3]

Substituting these values in 2.1, we find: [2.4]

Limiting Values

Nonnegative Integers

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 nx^{n-1}.

Rational Numbers

Let n=p/q, where p and q are nonnegative integers. Then [3.2]

Rearranging 3.2: [3.3]

Writing x^{1/q}=x and y^{1/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=x^{1/q}: [3.6]

So, once again, the limit is nx^{n-1}.

Negative Numbers

Let n=p, where p is a positive integer, so: [3.7]

Rearranging: [3.8]

Multiplying by -1 to bring the right-hand side into our normal form, with a positve x^{p}: [3.9]

Taking the limit, and noting 1/x^{2p}=x^{-2p}: [3.10]

Re-substituting n=-p [3.11] So, once again, the limit is nx^{n-1}. Therefore, we conclude that [3.12]

Where
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).

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