Question

Find a power series for the function, centered at \(c,\) and determine the interval of convergence. $$ f(x)=\frac{1}{2-x}, \quad c=5 $$

Short answer

The power series of the function \(f(x)=\frac{1}{2-x}\), centered at \(c = 5\), is \(f(x) = \sum_{n=0}^{\infty} (7 - x)^n\). The series converges on the interval \(4 < x < 6\).

Step-by-step solution

Center the Function

Since we want to find the power series centered at \(c = 5\) instead of \(c = 0\), adjust the function so that it is in the form of \(\frac{1}{1 - (x-c)}\). In this case, the function becomes \(f(x)=\frac{1}{2-(x - 5)} = \frac{1}{7 - x}\).

Find the Power Series

Recall that the standard power series for the function \(\frac{1}{1-x}\) is \( \sum_{n=0}^{\infty} x^n \).We can use this, with \(x\) replaced by \(7 - x\), to write the power series for our function. The power series becomes \(f(x) = \sum_{n=0}^{\infty} (7 - x)^n\).

Determine the Radius of Convergence

The radius of convergence \( R \) for the series \( \sum_{n=0}^{\infty} x^n \) is 1. Thus, \( R \) for our series is also 1.

Determine the Interval of Convergence

The interval of convergence for the standard power series about \(c = 5\), is from \(c - R = (5 - 1)\) to \(c + R = (5 + 1)\). Thus, the interval of convergence for our series is from 4 to 6.