Power Series Question 1

The Maclaurin power-series for the function $f$ is given by $\displaystyle f(x)=\sum_{n=0}^\infty \left(-\frac{x}{4} \right)^n$ . What is the value of $f(3)$ ?

-3

-\dfrac{3}{7}

\dfrac{4}{7}

-\dfrac{13}{16}

Approach

The power-series represents the sum of the geometric power-series with starting value $a_1=1$ and ratio $-\dfrac{x}{4}$ :

S_n= \frac{a_1}{1-r}=\frac{1}{1-\left(-\frac{x}{4}\right)}

f(3) = \frac{1}{1-\left(-\frac{3}{4}\right)}

= \frac{1}{\frac{7}{4}}

= \boxed{\frac{4}{7}}