Determine the power series representation of the function $f(x)=\sqrt{(4+x)^3}$ and indicate the radius of convergence I want to find a representation of the function mentioned above, so I took into account that:
$$f(x)=\sqrt{(4+x)^3}=8\left(1+\frac{x}{4}\right)^{\frac{3}{2}}$$ and developing the binomial series for $\left(1+\frac{x}{4}\right)^{\frac{3}{2}}$ we have to
$$\left(1+\frac{x}{4}\right)^{\frac{3}{2}}=1+\frac{3}{2}\left(\frac{x}{4}\right)+\frac{\frac{3}{4}}{2!}\left(\frac{x}{4}\right)^2+\frac{\frac{-3}{8}}{3!}\left(\frac{x}{4}\right)^3+\cdot\cdot\cdot$$
Now, to obtain the terms of the original series simply multiply by 8; now, how can I express the function as a series, since I have only managed to express that as a sum of terms
Any help is appreciated

According to the binomial series expansion we obtain
\begin{align*}
\left(1+\frac{x}{4}\right)^{\frac{3}{2}}&=1+\frac{3}{2}\left(\frac{x}{4}\right)+\frac{\frac{3}{4}}{2!}\left(\frac{x}{4}\right)^2+\frac{\frac{-3}{8}}{3!}\left(\frac{x}{4}\right)^3+\cdots\\
&=1+\binom{3/2}{1}\frac{x}{4}+\binom{3/2}{2}\left(\frac{x}{4}\right)^2+\binom{3/2}{3}\left(\frac{x}{4}\right)^3+\cdots\\
&\,\,\color{blue}{=\sum_{j=0}^\infty\binom{3/2}{j}\left(\frac{x}{4}\right)^j}
\end{align*}
