Find the first four terms of the binomial series for the function (1 + 8 x)^{1 / 2}.

Question:

Find the first four terms of the binomial series for the function {eq}\displaystyle (1 + 8 x)^{\dfrac 1 2} {/eq}.

Series for Fractional Power:

The mathematical expression that we are going to use in this problem for the function {eq}(1+x)^m {/eq} is shown below:

{eq}\displaystyle (1+x)^m=1+mx+\frac{m(m-1)}{2!}x^2+\frac{m(m-1)(m-2)}{3!}x^3+\dots {/eq}, where,

• m represents the value of the fractional power of the given algebraic function.

The given expression with fractional power is:

{eq}\displaystyle f(x)= (1 + 8 x)^{\dfrac 1 2} {/eq}

After comparision of the above function with the expression {eq}(1+x)^{m} {/eq}, we have:

{eq}m=\dfrac 1 2\\ x= 8 x {/eq}

Simplifying the coefficients of the binomial series formula with the above values, we have:

{eq}\begin{align*} \displaystyle (1 + 8 x)^{\dfrac 1 2}&=1+\frac{1}{2}(8 x)+\frac{\frac{1}{2}\left ( \frac{1}{2}-1 \right )}{2!}(8 x)^2+\frac{\frac{1}{2}\left ( \frac{1}{2}-1 \right )\left ( \frac{1}{2}-2 \right )}{3!}(8 x)^3+\dots\\ &=1+4x+\frac{\frac{1}{2}\left (- \frac{1}{2}\right )}{2}(64 x^2)+\frac{\frac{1}{2}\left (- \frac{1}{2} \right )\left ( -\frac{3}{2} \right )}{3(2)(1)}(512 x^3)+\dots&\because (xy)^a=x^a\cdot y^a\\ &=1+4x-\frac{1}{8}(64 x^2)+\frac{1}{16}(512 x^3)+\dots\\ &=\boxed{1+4x-8 x^2+32x^3+\dots} \end{align*} {/eq}