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Ken Ward's Mathematics Pages

Series Binomial 3

Series Contents

Page Contents

  1. Formula
  2. Proof
  3. Limiting Values
    1. n is nonnegative
    2. n is a rational number
    3. n is a negative integer

Formula

The following formula is sometimes called Binomial 3. It is the difference between two different numbers raised to the same power binomial3Formul.gif [1.1]
Or
binomial36.gif

Examples are:
binomial31.gif [1.2]

binomial32 [1.3]
binomial33.gif [1.4]

Proof

Assume binomial3Assumption.gif[2.1]
(Method of Undetermiate Coefficients)

The a's are numbers independent of  x and y.

Multiply both sides by (x-y):
binomial34.gif [2.2]

Note: binomial35.gif, and is just a convenient way of writing he terms. For instance the coefficient of y is a2x-a1

Equating coefficients, we find:
binomial3EquatedCoefficients.gif [2.3]


Substituting these values in 2.1, we find:
binomial36.gif[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:binomial3limitPos.gif [3.1]
Noting that the limit is nxn-1.

Rational Numbers

Let n=p/q, where p and q are nonnegative integers. Then
binomial37.gif [3.2]

Rearranging 3.2:
binomial38.gif  [3.3]

Writing x1/q=x and y1/q=y. Also dividing top and bottom by (x-y):
binomial39.gif [3.4]

Taking the limit (recalling that p is a positive integer):
binomial310.gif [3.5]

Reverting to n=p/q and x=x1/q:
binomial311.gif [3.6]

So, once again, the limit is nxn-1.

Negative Numbers

Let n=p, where p is a positive integer, so:binomial312.gif [3.7]

Rearranging:
binomial313.gif [3.8]

Multiplying by -1 to bring the right-hand side into our normal form, with a positve xp:
binomial314.gif [3.9]

Taking the limit, and noting 1/x2p=x-2p:
binomial315.gif [3.10]

Re-substituting n=-p
binomial316.gif [3.11]
So, once again, the limit is nxn-1. Therefore, we conclude that
binomial317.gif [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).



Ken Ward's Mathematics Pages


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