Find a power series for the given function. $f ( x ) = \frac { x } { 2 - x }$
A) $\sum _ { n = 0 } ^ { \infty } \left( \frac { x } { 2 } \right) ^ { n }$
B) $\sum _ { n = 0 } ^ { \infty } \frac { x ^ { n + 1 } } { 2 }$
C) $\sum _ { n = 0 } ^ { \infty } \frac { x ^ { n + 1 } } { 2 ^ { n + 1 } }$
D) $\sum _ { n = 0 } ^ { \infty } \frac { x ^ { n + 1 } } { 2 ^ { n } }$

Question

Quizplus · Accepted Answer

The function $f(x) = \frac{x}{2-x}$ can be rewritten as $f(x) = \frac{x}{2(1-\frac{x}{2})}$. Recognizing the geometric series formula $\sum_{n=0}^{\infty} ar^n = \frac{a}{1-r}$ where $a = x$ and $r = \frac{x}{2}$, the power series expansion is $\sum_{n=0}^{\infty} \frac{x^{n+1}}{2^{n+1}}$, matching choice C.

Find a Power Series for the Given Function f(x)=x2−xf ( x ) = \frac { x } { 2 - x }f(x)=2−xx​

Find a Power Series for the Given Function $f ( x ) = \frac { x } { 2 - x }$