$\text { Find a power series for the function } \frac { 12 } { 12 x + 13 } \text { centered at } 0 \text {. }$
A) $\sum _ { n = 0 } ^ { \infty } - \left( \frac { 12 } { 13 } \right) ^ { n + 1 } x ^ { n }$
B)
$\sum _ { n = 0 } ^ { \infty } ( - 1 ) ^ { n } \left( \frac { 12 } { 13 } \right) ^ { n + 1 } x ^ { n }$
C)
$\sum _ { n = 0 } ^ { \infty } \left( \frac { 12 } { 13 } \right) ^ { n + 1 } x ^ { n }$
D)
$\sum _ { n = 0 } ^ { \infty } ( - 1 ) ^ { n } \left( \frac { 12 x } { 13 } \right) ^ { n + 1 }$
E)
$\sum _ { n = 0 } ^ { \infty } ( - 1 ) ^ { n } \left( \frac { 12 x } { 13 } \right) ^ { n } + 1$

Question

Quizplus · Accepted Answer

We can use the formula $\frac{1}{1-x} = \sum_{n=0}^\infty x^n$ for $|x|<1$. We can rewrite the given function as $\frac{12}{13}\cdot \frac{1}{1+\frac{12}{13}x}$. Then, using the formula with $x' = -\frac{12}{13}x$, we have
$$\frac{12}{13}\cdot \frac{1}{1+\frac{12}{13}x} = \frac{12}{13} \cdot \sum_{n=0}^\infty \left(-\frac{12}{13} x\right)^n = \sum_{n=0}^\infty (-1)^n \cdot \left(\frac{12}{13}\right)^{n+1} x^n.$$
Thus, the answer is $\boxed{\textbf{(B)}}$.

Find a power series for the function 1212x+13 centered at 0. \text { Find a power series for the function } \frac { 12 } { 12 x + 13 } \text { centered at } 0 \text {. } Find a power series for the function 12x+1312​ centered at 0.

$\text { Find a power series for the function } \frac { 12 } { 12 x + 13 } \text { centered at } 0 \text {. }$