Find a power series representation for
$$f ( t ) = \ln ( 14 - t )$$
A)
$\ln 14 - \sum _ { n = 1 } ^ { \infty } \frac { t ^ { n } } { 14 ^ { n } }$
B)
$\ln 14 - \sum _ { n = 1 } ^ { \infty } \frac { t ^ { n } } { n 14 ^ { n } }$
C)
$\sum _ { n = 0 } ^ { \infty } \frac { t ^ { n } } { n 14 ^ { n } }$
D)
$\sum _ { n = 1 } ^ { \infty } \frac { 14 t ^ { n } } { n ^ { n } }$
E)
$\ln 14 + \sum _ { n = 1 } ^ { \infty } \frac { t ^ { 2 n } } { 14 ^ { n } }$

Question

Quizplus · Accepted Answer

We can use the Taylor series expansion of $\ln(1-x)$ centered at $x=0$, which is 
$$\ln(1-x)=-\sum_{n=1}^{\infty} \frac{x^n}{n}$$
We can then substitute $t=14-x$ and get 
$$\ln(14-t)=-\ln(1-\frac{t}{14})=\sum_{n=1}^{\infty} \frac{t^n}{n14^n}$$
Thus the correct choice is $\boxed{\textbf{(B)}}$.

Find a Power Series Representation For f(t)=ln⁡(14−t)f ( t ) = \ln ( 14 - t )f(t)=ln(14−t)

Find a Power Series Representation For $f ( t ) = \ln ( 14 - t )$