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An infinite series is a sum of infinitely many terms. The sequence of terms in our series starts with a term for which the index (often denoted by "n") is zero and continues without end. In this context, we are specifically dealing with a geometric series which is a type of infinite series. This means our series continues:

• The first term is: \((1 - x)^0 = 1\)
• The second term is: \((1 - x)^1\)
• The third term is: \((1 - x)^2\)
• And so on...

When looking at an infinite series, one important aspect to consider is whether the series converges—that is, does it approach a finite limit as the number of terms goes to infinity? This leads us to discuss series convergence in the next section.

Series convergence occurs when the sum of an infinite series settles to a particular value as the number of terms grows indefinitely. For a geometric series, convergence is contingent upon the absolute value of the series' common ratio. In our exercise, the common ratio is \( r = 1 - x \).To determine if an infinite geometric series converges, we must satisfy the condition:

• \( |r| < 1 \)

For our specific series \(\sum_{n=0}^{\infty}(1-x)^{n}\), this means we need:

• \(|1 - x| < 1\)

Solving this absolute inequality gives us the convergence interval for \( x \):

• When \(|1 - x| < 1\), the inequality translates to: \(0 < x < 2\)

This tells us the series converges only for values of \(x\) within this range. By ensuring these conditions, we lay the groundwork for determining the exact sum, or closed-form expression, of our series, as detailed in the upcoming section.

Finding a closed-form expression means deriving a finite formula that represents the sum of an infinite series without endlessly adding each term. For infinite geometric series, the closed-form expression is often derived using the sum formula:\[ S = \frac{a}{1-r} \]where \(a\) is the first term and \(r\) is the common ratio. In our case:

• \(a = 1\)
• \(r = 1 - x\)

By inserting these into our formula, we compute the sum as:\[ S = \frac{1}{1 - (1-x)} = \frac{1}{x} \]This result is significant because it provides a simple and immediate way to compute the sum of our series given that \(x\) is in the convergence interval \(0 < x < 2\). It's a powerful example of how infinite processes can yield simple, elegant solutions.

The common ratio in a geometric series is the factor by which each term is multiplied to get to the next term. It's a key element in determining both the behavior and convergence of the series.In our given series \(\sum_{n=0}^{\infty}(1-x)^{n}\), the common ratio is observed through the form of each term relative to its predecessor:

• The first term: \((1-x)^0 = 1\)
• The second term: \((1-x)^1\)
• The third term: \((1-x)^2\)

These terms reflect the relationship:

• The second term is the first multiplied by \(1-x\).
• The third term is the second term multiplied again by \(1-x\), and so forth.

Thus, the common ratio is \(1 - x\). It is vital because:

• If \(|1-x| < 1\), the series converges.
• Otherwise, it diverges, meaning it doesn't settle to any finite value.

Understanding the common ratio helps in grasping both the exponential progression of series terms and the convergence criteria, solidifying your capability to explore geometric series analytically.