Consider the function given by $f ( x ) = \sum _ { n = 0 } ^ { \infty } \left( \frac { x } { 14 } \right) ^ { n }$.
Find the interval of convergence for $f ^ { t } ( x )$
A) $[ - 14,14 )$
B) $( - 14,0 ]$
C) $( 0,14 )$
D) $[ - 14,14 ]$
E) $( - 14,14 )$

Question

Quizplus · Accepted Answer

We use the ratio test to find the interval of convergence for $f ^ { t } ( x )$:
$$\lim_{n\rightarrow\infty}\left|\frac{(x/14)^{n+1}}{(x/14)^n}\right|=\lim_{n\rightarrow\infty}\left|\frac{x}{14}\right|=|x/14|$$
The series converges if $|x/14|<1$ or $-14<x<14$. Therefore, the interval of convergence for $f ^ { t } ( x )$ is $( - 14,14 )$. So the answer is E.

Consider the Function Given By f(x)=∑n=0∞(x14)nf ( x ) = \sum _ { n = 0 } ^ { \infty } \left( \frac { x } { 14 } \right) ^ { n }f(x)=∑n=0∞​(14x​)n

Consider the Function Given By $f ( x ) = \sum _ { n = 0 } ^ { \infty } \left( \frac { x } { 14 } \right) ^ { n }$