An expression of the form:
\[ (1 + x)^{n} \]
where \(|x| < 1\) and \(n\) is a non-integer, can be approximated using sufficiently many terms of the expression:
\[ 1 + nx + n(n-1)\frac{x^{2}}{2!} + n(n-1)(n-2)\frac{x^{3}}{3!} + ... \]
| Expression | \(1\) | \(x\) | \(x^{2}\) | \(x^{3}\) |
|---|---|---|---|---|
| \((1 + x)^{\frac{1}{2}}\) | \(1\) | \(\frac{1}{2}\) | \(-\frac{1}{8}\) | \(\frac{1}{16}\) |
| \(1 - \sqrt{1 - x}\) | \(0\) | \(\frac{1}{2}\) | \(-\frac{1}{8}\) | \(\frac{1}{16}\) |
| \((1+x)^{-\frac{1}{2}}\) | \(1\) | \(-\frac{1}{2}\) | \(\frac{3}{8}\) | \(-\frac{5}{16}\) |
| \((1 - 2x^{2})^{-\frac{1}{2}}\) | \(1\) | \(0\) | \(1\) | \(0\) |