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]
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 xk approaches infinity, if xk becomes infinite. We therefore require that |x|<1, when the series is infinite.
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 (ak-bk)xk=0 for all values of x, when ak=bk, and, for an infinite series, (|x|<1).