P4 4.1 - Binomial series with rational powers

General binomial coefficients

Can a polynomial give a useful estimate of $\sqrt{1.04}$? An integer power such as $(1+x)^3$ expands to a finite polynomial. For a fractional or negative power, the same coefficient pattern continues indefinitely.

The first coefficient is $1$. Multiply it by $n$ for the coefficient of $x$, then by $(n-1)/2$ for the coefficient of $x^2$, then by $(n-2)/3$ for the coefficient of $x^3$. For a nonnegative integer $n$, a factor eventually becomes zero and the expansion terminates. That does not happen for a negative integer or a non-integer rational power.

Put negative or fractional values of $n$ in brackets while calculating coefficients. The factorial denominators remain $2!=2$, $3!=6$.

Change only the sign of the exponent. A reciprocal square root has a different sign pattern; it is not the negative of the first expansion.

A finite list of terms must retain an indication that more terms follow.

Keep the fractional factors visible while finding the next coefficients.

Taking out a constant factor

The binomial formula starts with $1+u$, so a different constant inside the bracket must be factored out before expanding.

For example, $16+x=16(1+x/16)$. Raising the whole product to $-1/2$ gives $16^{-1/2}(1+x/16)^{-1/2}$, so the outside factor is $1/4$, not $16$.

This also explains why a change of sign inside the bracket changes odd powers but leaves even powers unchanged.

Change only $x$ to $-x$. Use the existing series rather than recalculating all its coefficients.

The outside factor applies to the whole series.

Here the extracted constant has a different square root, but the same expansion method applies.

The interval of validity

A real function can exist at a value where its binomial series about zero is not valid. Always apply the condition to the expression taking the place of $u$, rather than automatically writing $|x|<1$.

For $(b+ax)^n$ with $a\ne0$ and $b>0$, the standard condition $|ax/b|<1$ becomes $|x|<b/|a|$. This is the open interval centred at zero. We use that strict condition here; it makes no claim about individual endpoint convergence.

[DIAGRAM: asset_name: Function domain and series validity; asset_slug: series-validity; caption: The function can exist beyond the interval in which its series about zero is valid.; description: Two aligned number lines for (9+x) to the power negative one half. The function domain is x greater than negative nine, while binomial series validity is negative nine less than x less than nine. Open circles mark excluded endpoints.]