Question

How do you use the binomial series to expand ${\left(1+x\right)}^{\frac{1}{2}}?$

Answer 1

Use binomial expansion to evaluate the expression:

From binomial series expansion,

${\left(1+x\right)}^n=1+nx+\frac{n(n-1)}{2}{x}^2+\frac{n(n-1)(n-2)}{6}{x}^3+......$

In this case, $n=\frac{1}{2}$.

Substituting the values,

${\left(1+x\right)}^{\frac{1}{2}}=1+\frac{1}{2}x-\frac{1}{4}{x}^2+\frac{1}{16}{x}^3+.....$.

Hence, the binomial series expansion can be given as ${\left(1+x\right)}^{\frac{\mathbf{1}}{\mathbf{2}}}\mathbf{=}\mathbf{1}\mathbf{+}\frac{\mathbf{1}}{\mathbf{2}}\mathit{x}\mathbf{-}\frac{\mathbf{1}}{\mathbf{4}}{\mathit{x}}^{\mathbf{2}}\mathbf{+}\frac{\mathbf{1}}{\mathbf{16}}{\mathit{x}}^{\mathbf{3}}\mathbf{+}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}$.