Consider the following series. Answer the following questions. 1. Find the values of \( x \) for which the series converges. Answer (in interval notation): 2. Find the sum of the series for those values of \( x \). Write the formula in terms of \( x \). Sum:

To determine the values of \( x \) for which the series converges, we'll typically look for a power series or a geometric series in the format of \( \sum a_n (x - c)^n \) or a similar structure. 1. The convergence of a geometric series \( \sum ar^n \) occurs when \( |r| < 1 \). If the series in question has a common ratio \( r(x) \), we set up the inequality \( |r(x)| < 1 \). Assuming you have a series like \( \sum_{n=0}^{\infty} a_n (x - c)^n \), you can then often find the radius of convergence \( R \) using the ratio test, leading you to an interval for \( x \). The answer will depend on the specifics of the series provided. For a generic series like \( \sum_{n=0}^{\infty} x^n \), convergence occurs for \( |x| < 1 \), hence the answer will be \( (-1, 1) \) in interval notation. 2. For the sum of a geometric series when it converges, if we've established that the series converges for \( |x| < 1 \), the formula for the sum of a geometric series is \( S = \frac{a}{1 - r} \). If our series is \( \sum_{n=0}^{\infty} x^n \), the first term \( a = 1 \) (for \( n=0 \)), and \( r = x \). Thus, the sum of this series can be expressed as \( S(x) = \frac{1}{1 - x} \) for \( |x| < 1 \).