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 know that the power series representation of ln(1+x) is:
$$\sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{n}x^n$$
By replacing x with -t/14 and multiplying the series by -1, we get:
$$\ln(14-t)=\ln 14 + \sum_{n=1}^{\infty}\frac{t^n}{n14^n}$$
This matches choice B. Therefore, the correct answer is 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 $f ( t ) = \ln ( 14 - t )$