Rational binomial expansions and validity — 9709 P3
Expand around zero and retain the strict validity interval.
Notes and key methods
For a rational exponent r, the expansion begins \(\displaystyle (1+ u)^{r}= 1+ ru+ \frac{r(r- 1)u^{2}}{2}+ \frac{r(r- 1)(r- 2)u^{3}}{6}+ \cdots\). The infinite binomial series is used for |u|<1.
Put \(\displaystyle u=ax\) before collecting coefficients. The validity condition becomes |ax|<1. An approximation uses only the retained terms, so write an approximation sign and check that its input is inside the interval.
Check the conditions
Expand about \(\displaystyle x=0\).
Use the strict interval |ax|<1 for the infinite series.
Three worked examples
Example 1 · Rational binomial expansions
Consider \(\displaystyle \frac{1}{\left(5 x + 1\right)^{2}}\) expanded in ascending powers of x.
Write \(\displaystyle a=5\) and \(\displaystyle r=\)\(-2\).
Expand through \(\displaystyle x^{3}\).
State the interval where this binomial series is valid.
Use the cubic truncation to estimate the expression at \(\displaystyle x= \frac{1}{10a}\).
Find the positive radius of the validity interval.
Show the reasoning
(a)
\(- 500 x^{3} + 75 x^{2} - 10 x + 1\) follows by substituting \(\displaystyle u=ax\) into the rational binomial series.
Here \(\displaystyle ax= \frac{1}{10}\) is inside the validity interval. Substitute into the cubic polynomial; the omitted higher terms make the value an approximation.
Answer: \(\displaystyle \frac{413}{500}\) approximately
(d)
Solve |ax|<1 for x.
\(\displaystyle \frac{1}{5}\)
Answer: \(\displaystyle \frac{1}{5}\).
Example 2 · Rational binomial expansions
Consider \(\displaystyle \left(6 x + 1\right)^{\frac{3}{2}}\) expanded in ascending powers of x.
Write \(\displaystyle a=6\) and \(\displaystyle r=\)\(\displaystyle \frac{3}{2}\).
Expand through \(\displaystyle x^{3}\).
State the interval where this binomial series is valid.
Use the cubic truncation to estimate the expression at \(\displaystyle x= \frac{1}{10a}\).
Find the positive radius of the validity interval.
Show the reasoning
(a)
\(\displaystyle - \frac{27 x^{3}}{2} + \frac{27 x^{2}}{2} + 9 x + 1\) follows by substituting \(\displaystyle u=ax\) into the rational binomial series.
Here \(\displaystyle ax= \frac{1}{10}\) is inside the validity interval. Substitute into the cubic polynomial; the omitted higher terms make the value an approximation.
Answer: \(\displaystyle \frac{18459}{16000}\) approximately
(d)
Solve |ax|<1 for x.
\(\displaystyle \frac{1}{6}\)
Answer: \(\displaystyle \frac{1}{6}\).
Example 3 · Rational binomial expansions
Consider \(\sqrt{2 x + 1}\) expanded in ascending powers of x.
Write \(\displaystyle a=2\) and \(\displaystyle r=\)\(\displaystyle \frac{1}{2}\).
Expand through \(\displaystyle x^{3}\).
State the interval where this binomial series is valid.
Use the cubic truncation to estimate the expression at \(\displaystyle x= \frac{1}{10a}\).
Find the positive radius of the validity interval.
Show the reasoning
(a)
\(\displaystyle \frac{x^{3}}{2} - \frac{x^{2}}{2} + x + 1\) follows by substituting \(\displaystyle u=ax\) into the rational binomial series.
Here \(\displaystyle ax= \frac{1}{10}\) is inside the validity interval. Substitute into the cubic polynomial; the omitted higher terms make the value an approximation.
Answer: \(\displaystyle \frac{16781}{16000}\) approximately
(d)
Solve |ax|<1 for x.
\(\displaystyle \frac{1}{2}\)
Answer: \(\displaystyle \frac{1}{2}\).
Common mistakes
Do not use the finite positive-integer restriction rule here.
Remember each power of a when substituting \(\displaystyle u=ax\).