Find the first four terms of the binomial series for the given function.
-$$\left( 1 + \frac { x } { 4 } \right) ^ { 3 }$$
A) $1 - \frac { 3 } { 4 } x + \frac { 3 } { 16 } x ^ { 2 } - \frac { 1 } { 64 } x ^ { 3 }$
B) $1 + \frac { 3 } { 4 } x + \frac { 3 } { 16 } x ^ { 2 } + \frac { 1 } { 64 } x ^ { 3 }$
C) $1 + \frac { 3 } { 8 } x + \frac { 3 } { 64 } x ^ { 2 } + \frac { 1 } { 512 } x ^ { 3 }$
D) $1 - \frac { 3 } { 8 } x + \frac { 3 } { 64 } x ^ { 2 } - \frac { 1 } { 512 } x ^ { 3 }$

Question

Quizplus · Accepted Answer

The binomial series expansion of $(1 + x)^n$ is given by $1 + nx + \frac{n(n-1)}{2!}x^2 + \frac{n(n-1)(n-2)}{3!}x^3 + \cdots$. For $\left(1 + \frac{x}{4}\right)^3$, substituting $n=3$ and $x$ with $\frac{x}{4}$, the first four terms are $1 + 3\left(\frac{x}{4}\right) + \frac{3(2)}{2!}\left(\frac{x}{4}\right)^2 + \frac{3(2)(1)}{3!}\left(\frac{x}{4}\right)^3$, simplifying to $1 + \frac{3}{4}x + \frac{3}{16}x^2 + \frac{1}{64}x^3$.

Find the First Four Terms of the Binomial Series for the Given