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Series Binomial Theorem Proof Using Algebra

Series Contents

Page Contents

  1. A Proof Using Algebra
  2. Behaviour of the series

A Proof Using Algebra

The following is a proof of the Binomial Theorem for all values, claiming to be algebraic. Because we use limits, it could be claimed to be another calculus proof in disguise.

We assume:
binomialalgebra1.gif [1.1]
That is, we assume the Principle of Indeterminate Coefficients, that (1+x)n is a polynomial of the stated kind. The coefficients of x are assumed to be a function of n, and not a function of x.

We use a number y instead of x:
binomialalgebra2.gif [1.2]
And the same result should follow.

Subtracting 1.2 from 1.1
binomialalgebra3.gif [1.3]

Divide the left-hand side by (1+x)-(1+y), and the right-hand side by (x-y), its equal:
binomialalgebra4.gif [1.4]

We now use limits as related to binomial 3.
binomial317.gif [1.5]

Using this formula, we observe, in 1.4 that:

binomialalgebra6.gif [1.7]

Multiplying throughout in 1.7 by (1+x):
binomialalgebra7.gif [1.8]

Multiplying 1.1 by n:
binomialalgebra8.gif [1.9]

Equating coefficients in 1.8 and 1.9:

 binomialalgebra10.gif [1.11]

binomialalgebra11.gif [1.12]

binomialalgebra.gif [1.13]

And, in general:
binomialalgebra14.gif [1.13.1]

Substituting these values in 1.1, and taking out the common a0 factor:
binomialalgebra12.gif [1.14]
When x=0:
binomialalgebra13.gif [1.15]

We take the arithmetic root of 1, which gives a0=1, but we note that there are actually n roots of 1, and we have arbitrarily chosen 1.

We therefore conclude, that for all rational numbers, r:
binomialAlgebraTheorem.gif [1.15]

Where r is a rational number.

Behaviour of the series

The following is a formula for the relationship between binomial coefficients:binomialAlgebraRecursiveFormula.gif [2.1]

When r is a non-negative integer, the terms will become zero when k=r+1.

When r is not a non-negative integer, then, after k exceeds r, the subsequent terms will alternate in sign, each one being the negative of the previous one, and the series will continue infinitely.

The validity of our argument depends on the validity of the Principle of Indeterminate Coefficients, which depends on x not being infinite. As the x-value of the kth term is xk, then for the terms to remain finite as k approaches infinity, |x|<1, when r is a not a non-negative integer. The value |x|<1 ensures the terms approach zero as k tends to infinity (which is what we need for the Principle of Indeterminate Coefficients) and it means the series could be convergent (which it actually is), but we have not proved this, yet.

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