Which of the following is the power series centered at $a = 0$ for $f ( x ) = \frac { 1 } { 1 - 4 x ^ { 2 } }$ ?
1) $\sum _ { n = 0 } ^ { \infty } 4 ^ { n } x ^ { 2 n }$ 2) $\sum _ { n = 0 } ^ { \infty } ( 2 x ) ^ { 2 n }$ 3) $\sum _ { n = 0 } ^ { \infty } ( - 1 ) ^ { n } 4 ^ { n } x ^ { 2 n }$
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 $f ( x ) = \frac { 1 } { 1 - 4 x ^ { 2 } }$ ?
1) $\sum _ { n = 0 } ^ { \infty } 4 ^ { n } x ^ { 2 n }$ 2) $\sum _ { n = 0 } ^ { \infty } ( 2 x ) ^ { 2 n }$ 3) $\sum _ { n = 0 } ^ { \infty } ( - 1 ) ^ { n } 4 ^ { n } x ^ { 2 n }$ 
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 power series for $f(x) = \frac{1}{1 - 4x^2}$ can be found by recognizing it as a geometric series with the first term $a = 1$ and common ratio $r = 4x^2$. The formula for a geometric series is $\sum_{n=0}^{\infty} ar^n$, which in this case translates to $\sum_{n=0}^{\infty} (4x^2)^n$. This can be expressed in two ways: as $\sum_{n=0}^{\infty} 4^n x^{2n}$ and as $\sum_{n=0}^{\infty} (2x)^{2n}$, making both choices 1 and 2 correct. Choice 3 introduces a factor of $(-1)^n$, which does not match the given function's series expansion.

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