Question

Define a power series centered at \(c\).

Short answer

A power series centered at \( c \) is defined by \( \sum_{n=0}^{∞} c_n (x - c)^n \)

Step-by-step solution

Define a general power series

Firstly, we need to understand a general power series. A power series is an infinite series of the form \( \sum_{n=0}^{∞} c_n (x - a)^n \), where \( c_n \) are the coefficients of the series, \( x \) is the variable, and \( a \) is the center of the series.

Center the power series at c

To center a power series at any point \( c \), we replace \( a \) with \( c \) in the general power series. This gives us the power series centered at \( c \), which is defined by \( \sum_{n=0}^{∞} c_n (x - c)^n \)