Binomial Series — Question 9

*Question 9 Find the coefficient of $x^4$ in the Maclaurin series for $(1+x)^{1/2}$ directly from the generalized binomial formula. Do not compute the lower-degree coefficients.

*Question 9 – Solution

In $$(1+x)^{1/2}=\sum_{n=0}^{\infty}\binom{1/2}{n}x^n,$$ the coefficient of $x^4$ is exactly $\binom{1/2}{4}$.

$$\begin{align*} \binom{1/2}{4}&=\frac{(1/2)(1/2-1)(1/2-2)(1/2-3)}{4!}\\&=\frac{(1/2)(-1/2)(-3/2)(-5/2)}{24}\\&=\frac{-15}{16\cdot24}=\boxed{-\frac5{128}}. \end{align*}$$ There are three negative numerator factors, so the negative sign is expected.

The $x^4$ term is $\boxed{-5x^4/128}$. Computing preceding terms is unnecessary because generalized binomial coefficients directly target any desired power.