Question

In Exercises 1 and 2 , state where the power series is centered. $$ \sum_{n=0}^{\infty} n x^{n} $$

Short answer

The power series is centered at \(x=0\).

Step-by-step solution

Understand Power series

Recognize that a power series takes the form: \(\sum_{n=0}^{\infty} a_n (x-c)^n\). In our series \(\sum_{n=0}^{\infty} n x^{n}\), the \(a_n\) term is \(n\), and all \(x\) terms are in the form \(x^n\), which can be rewritten as \((x-0)^n\). This indicates that the series matches the general form of a power series.

Identify the center

Based on comparison with the general form of a power series, where \(c\) is the center of the series, in \(\sum_{n=0}^{\infty} n (x-c)^{n}\), it can be seen that the \(x\) term in our power series is raised to the \(n\) power as \(x^n\), or \((x-0)^n\). Hence, \(c=0\) is the center of the power series.