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Geometric series are a special type of series that are both simple and extremely useful in mathematics. They are characterized by each term being a constant multiple of the previous term. This is called the "common ratio." For a series to be classified as geometric, it must have the form:

• First term: \( a \)
• Common ratio: \( r \) for each successive term

In our example, the series given is \( \sum_{n=0}^{\infty}\left(\frac{x^{2}+1}{3}\right)^{n} \). Here, \( a = 1 \) and the common ratio \( r = \frac{x^{2}+1}{3} \). Each term is generated by multiplying the previous term by this common ratio \( r \). This is why understanding the concept of the common ratio is crucial. It effectively determines whether the series converges or diverges and at what values of \( x \) this happens.

To find the interval of convergence for a geometric series, we must solve an inequality related to the common ratio. For convergence, the absolute value of the common ratio, \( |r| \), must be less than 1. This ensures that as \( n \) increases, the terms of the series get smaller, leading them to converge to a finite sum.
The inequality for the given series is \( \left|\frac{x^{2}+1}{3}\right| < 1 \). To solve this, we remove the absolute value and solve the compound inequality:\(-3 < x^{2} + 1 < 3\). Subtracting 1 from each part, we get \(-4 < x^{2} < 2\). Considering that \( x^2 \) is always non-negative, this simplifies further to \( 0 \leq x^2 < 2\). Thus, the possible values for \( x \) lie in the interval \( -\sqrt{2} < x < \sqrt{2} \). This region indicates where the series converges, making it valid to use the sum formula.

Once the interval of convergence is found, we can use it to determine the sum of the series as a function of \( x \). For a geometric series \( \sum ar^n \) that converges, the sum \( S \) is given by the formula:\[S = \frac{a}{1-r}\]Let's apply this to the series \( \sum_{n=0}^{\infty}\left(\frac{x^{2}+1}{3}\right)^{n} \). Here, \( a = 1 \) and \( r = \frac{x^{2}+1}{3} \), so:\[S(x) = \frac{1}{1 - \left(\frac{x^{2}+1}{3}\right)} = \frac{3}{3 - (x^{2}+1)} = \frac{3}{2-x^{2}}\]This expression \( S(x) \) is valid only within the interval \( -\sqrt{2} < x < \sqrt{2} \).
This function of \( x \) provides the sum of the entire series and can be used in various analyses, from solving mathematical problems to modeling real-world scenarios that fit this geometric pattern.