(Solved): Find the Taylor series centered at \( c=4 \). \[ f(x)=3 \sqrt{x} \] Identify the correct expansion. ...

Find the Taylor series centered at \( c=4 \). \[ f(x)=3 \sqrt{x} \] Identify the correct expansion. \[ \begin{array}{lr} 3 \sum_{n=0}^{\infty}\left(\begin{array}{l} \frac{1}{2} \\ n \end{array}\right)(x-4)^{n} & 2 \sum_{n=0}^{\infty}\left(\begin{array}{l} \frac{1}{2} \\ n \end{array}\right)\left(\frac{x-4}{4}\right)^{n} \\ 3 \sum_{n=0}^{\infty}\left(\begin{array}{l} \frac{1}{2} \\ n \end{array}\right)\left(\frac{x-4}{4}\right)^{n} & 3 \sum_{n=0}^{\infty} \frac{1}{2^{2 n-1}}\left(\begin{array}{l} \frac{1}{2} \\ n \end{array}\right)(x-4)^{n} \end{array} \] Find the interval on which the expansion is valid. (Give your answer as an interval in the form \( \left(*^{*},\right) \). Use the symbol \( \infty \) for infinity, \( \cup \) for combining intervals, and an appropriate type of parenthesis "(".")", "["."]" depending on whether the interval is open or closed. Enter \( \oiint \) if the interval is empty. Express numbers in exact form. Use symbolic notation and fractions where needed.) interval: