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 start by using the formula for the natural logarithm: $\ln(1-x)=-\sum_{n=1}^\infty\frac{x^n}{n}$ for $|x|<1$. Then we can substitute $x=\frac{t}{14}$ to get $\ln\left(1-\frac{t}{14}\right)=-\sum_{n=1}^\infty\frac{t^n}{n14^n}$. Finally, we use the fact that $\ln(14-t)=-\ln\left(1-\frac{t}{14}\right)$ and rearrange to get $\ln(14-t)=\ln(14)-\sum_{n=1}^\infty\frac{t^n}{n14^n}$, which matches option 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 $f ( t ) = \ln ( 14 - t )$