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A geometric series is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. For example, in the series \(1, r, r^2, r^3, \ldots\), each term is the product of the common ratio and the term before it.

Now, when we talk about an infinite geometric series, we mean a series that continues forever. The sum of an infinite geometric series can be finite or infinite depending on the value of the common ratio. If the common ratio is between -1 and 1 (excluding -1 and 1), the series converges to a finite value; otherwise, it diverges, meaning it does not sum up to a finite value. In the exercise given, the series \( 1000+1055+1113.025+\ldots \) does not stop, implying it's an infinite geometric series.

The common ratio in a geometric series is what we multiply each term by to get the next term. It is a crucial element because it determines whether the series converges or diverges. We typically represent it with the variable \( r \).

The common ratio can be a whole number, a fraction, a decimal, or even a negative number. To find the common ratio in a geometric series, you divide any term by the previous term (except the first one). In the exercise, the common ratio is found by looking at the series formula: \(1000(1.055)^n\), which tells us that the common ratio is \(1.055\). This is greater than 1, which means as \(n\) increases, the terms of the series become larger and larger, leading to divergence.

Convergence and divergence are terms used to describe the behavior of infinite series. A series converges if the sum of its infinite terms approaches a finite value as the number of terms increases. Conversely, a series diverges if the sum tends to infinity as more and more terms are added.

For convergence in a geometric series, a necessary condition is that the absolute value of the common ratio \( \lvert r \rvert\) must be less than 1. In essence, the terms of the series get smaller and smaller, eventually adding up to a specific limit. However, if \( \lvert r \rvert \geq 1\), the terms either stay the same size or get larger as we move through the series, leading to divergence.

In the given exercise, since the common ratio is \(1.055\), we observe that the series does not have a sum that approaches a finite number. Hence, we say the series diverges because the condition for convergence is not met.