Which of the following is the power series centered at $a = 0$ for $f ( x ) = \frac { 1 } { 1 + 4 x }$ ?
1) $\sum _ { n = 0 } ^ { \infty } ( - 1 ) ^ { n } 4 ^ { n } x ^ { n }$ 2) $\sum _ { n = 0 } ^ { \infty } 4 ^ { n } x ^ { n }$ 3) $\sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ { n } } { 4 ^ { n } } x ^ { n }$
A)None
B)1
C)2
D)3
E)1, 2
F)1, 3
G)2, 3
H)1, 2, 3

Question

Which of the following is the power series centered at $a = 0$ for $f ( x ) = \frac { 1 } { 1 + 4 x }$ ?
1) $\sum _ { n = 0 } ^ { \infty } ( - 1 ) ^ { n } 4 ^ { n } x ^ { n }$ 2) $\sum _ { n = 0 } ^ { \infty } 4 ^ { n } x ^ { n }$ 3) $\sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ { n } } { 4 ^ { n } } x ^ { n }$ 
A)None
B)1
C)2
D)3
E)1, 2
F)1, 3
G)2, 3
H)1, 2, 3

Quizplus · Accepted Answer

The correct power series for $f(x) = \frac{1}{1 + 4x}$ is given by the geometric series formula, which in this case is $\sum_{n=0}^{\infty} (-1)^n 4^n x^n$, matching choice 1. This is because the function can be seen as a geometric series with a common ratio of $-4x$, which when expanded, matches the form of choice 1.

Which of the Following Is the Power Series Centered At a =