4 ### Geometric Series Convergence and Sum---**Problem Statement:**Consider the geometric series\[\sum_{n=0}^{\infty} \left( -\frac{1}{8} \right)^n (x - 7)^n.\]1. **For which \( x \)-values does the series converge?** * \( x \) is in [_____] (Enter your answer using [interval notation](https://en.wikipedia.org/wiki/Interval_(mathematics)).2. **For the \( x \)-values you found above, what is the sum of the series?** \[ \sum_{n=0}^{\infty} \left( -\frac{1}{8} \right)^n (x - 7)^n = [_____] \]---**Instructions:**- Identify the values of \( x \) for which the given series converges by using appropriate methods for determining the radius and interval of convergence for geometric series.- Find the closed-form expression for the sum of the series for those \( x \)-values within the interval of convergence.**Note:** The notation \(\sum_{n=0}^{\infty}\) represents an infinite sum starting from \( n = 0 \) to \( n = \infty \). The interval notation of the form [a, b) indicates an interval that is closed on the left and open on the right.---**Additional Help:**For detailed steps on determining the interval of convergence and calculating the sum of geometric series, refer to resources on [Geometric Series](https://en.wikipedia.org/wiki/Geometric_series) and [Series Convergence](https://en.wikipedia.org/wiki/Convergent_series).

Name: 4 ### Geometric Series Convergence and Sum---**Problem Statement:**Con
Uploaded: 2024-03-20T07:06:41.000Z
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Description: 4 ### Geometric Series Convergence and Sum---**Problem Statement:**Consider the geometric series\[\sum_{n=0}^{\infty} \left( -\frac{1}{8} \right)^n (x - 7)^n.\]1. **For which \( x \)-values does the series converge?** * \( x \) is in [_____] (Enter