Use the binomial series to find the maclaurin series for the function. f(x) = 1/(1 + x)⁴

Solution:

Binomial series can be written as

\((1+x)^{k}=1+kx+\frac{k(k-1)x^{2}}{2!}+\frac{k(k-1)(k-2)x^{3}}{3!}+.................\\Here, k = -4\\(1+x)^{-4}=1+(-4)x+\frac{(-4)(-4-1)x^{2}}{1\times 2}+\frac{(-4)(-4-1)(-4-2)x^{3}}{1\times 2\times 3}+.....\\(1+x)^{-4}=1-4x+\frac{20x^{2}}{2}-\frac{120x^{3}}{6}+....\\(1+x)^{-4}=1-4x+10x^{2}-20x^{3}+......\)

Therefore, the maclaurin series for the given function is \((1+x)^{-4}=1-4x+10x^{2}-20x^{3}+.....\)

Use the binomial series to find the maclaurin series for the function. f(x) = 1/(1 + x)⁴

Summary:

The maclaurin series for the function f(x) = 1/(1+x)⁴ using binomial series is \((1+x)^{-4}=1-4x+10x^{2}-20x^{3}+.....\)