State where the power series is centered.
A)0
B)-2
C)7
D)2
E)-7

Question

State where the power series  is centered. ​
A)0 
B)-2 
C)7 
D)2 
E)-7

Quizplus · Accepted Answer

The power series is of the form $\sum_{n=0}^{\infty} a_n(x-c)^n$, where $c$ is the center. Looking at the given series, we can see that the $(x-c)^n$ term is always positive (since it is being raised to an even power). Therefore, the sign of the coefficients will determine the sign of the entire series. We can see that as $n$ increases, the coefficients also increase, so the series will diverge for all $x$ except for $x=0$. Hence, the center of the series must be $0$, which corresponds to choice C.

State Where the Power Series Is Centered