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Arithmetic Algebra: Method of Indeterminate Coefficients

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Page Contents

1. Method of Indeterminate Coefficients (Method of Unknown Coefficients)
2. Proof

Method of Indeterminate Coefficients (Method of Unknown Coefficients)

Suppose we wish to find the expansion of:
[1.1]

(having forgotten the Binomial Theorem, etc.)

We assume:
[1.2]

Where the a's are independent of x.

Multiplying throughout by (1+x), we have:
[1.3]

Equating coefficients:

1=a0	Equating Constants
0=a0+a1, a1=-1	Equating powers of x
0=a1+a2, a2=1	Equating powers of x2
0=a2+a3, a3=-1	Equating powers of x3

Hence:

Proof

We argue that if a function, f(x):
[2.1]

Can be expressed as above, in terms of increasing powers of x, where k is a nonnegative integer, and the coefficients are independent of x, then, if we have a function, g(x):
[2.2]
Similarly, with increasing powers of x, with k as a nonnegative integer independent of x, then if f(x) and g(x) are identical, then there coefficients are identical. Further, that if the coefficients are identical, then so are f(x) and g(x).

If f(x) and g(x) are identical functions, then:
[2.3]
For all values of x.
Therefore, equating 2.1 and 2.2, we have:
[2.4]

Subtracting the right-hand side of 2.4 from the left and grouping the powers of x, we have:
[2.5]

If 2.5 is true for all values of x, then the coefficients of the x's are equal. Of course, 2.5 can be equal to zero for some values of x, or even an infinite number, but it is equal to zero for all values of x when the coefficients are equal.

When the series is finite, then this is true, because no finite number multiplied by zero is other than zero.

However, if 2.5 is an infinite series, the terms may not all be zero when multiplied by zero, as x^k approaches infinity, if x^k becomes infinite. We therefore require that |x|<1, when the series is infinite.

It is important for us to avoid being confused between the series (2.1) converging (which we do not demand) and the terms not increasing to infinity, which we do require.

We further claim, that if the coefficients of two functions are identical, then they are identical functions. When the two functions are finite, then this is evidently so. When they are infinite series, then we require |x|<1. We claim this because in 2.5 when the a's are equal to the corresponding b's, the equations are and must be equal for all values of x, because (a_k-b_k)x^k=0 for all values of x, when a^k=b^k, and, for an infinite series, (|x|<1). 

See also: Polynomial Argument.





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