Question

Find all values of \(x\) for which the series converges. For these values of \(x,\) write the sum of the series as a function of \(x\). $$ \sum_{n=0}^{\infty} 4\left(\frac{x-3}{4}\right)^{n} $$

Short answer

The series converges for \(x\) in the open interval (1, 5), and the sum of the series as a function of \(x\) is \(S=16/(5-x)\).

Step-by-step solution

Identify the common ratio

The power series presented is basically the geometric series where the common ratio, \(r=(x-3)/4\). The geometric series converges if and only if \(|r| < 1\).

Find the interval of convergence

To find the interval of convergence, we solve the following inequality \(|(x-3)/4| < 1\). By multiplying out the modulus and rearranging, this inequality simplifies to -1 < (x-3)/4 < 1 which further simplifies to 1 < x < 5. Therefore, the series converges for \(x\) in the open interval (1, 5).

Find the function representing the sum of series

The formula for the sum of a geometric series, when it converges, is \(S = a / (1 - r)\) where \(a\) is the first term and \(r\) is the common ratio. Here, \(a=4\) and \(r=(x-3)/4\), so the sum may be written as \(S = 4/(1 - (x-3)/4)\). This simplifies to \(S=16/(5-x)\).