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When we talk about the convergence of a series, we are referring to whether the series approaches a finite limit as the number of terms grows infinitely large. In a geometric series, each term after the first is obtained by multiplying the previous term by a fixed, non-zero number called the common ratio. If the series converges, it means that adding an infinite number of terms results in a finite sum.
To determine if a geometric series converges, you need to examine the common ratio, denoted by \( r \). For convergence, the absolute value of \( r \) must be less than 1, or \(|r| < 1\). When this condition is met, the sum of the infinite geometric series can be calculated using the formula:
\[ S = \frac{a}{1 - r} \]
where \( a \) is the first term of the series. If this condition is not satisfied, the series does not converge.

The common ratio is a crucial element in understanding geometric series. It is the constant factor between consecutive terms in the series. Identifying it helps determine the nature of the series, whether it converges or diverges.
To find the common ratio \( r \), you simply divide the second term of the series by the first term. For instance, in the series \( 1 + 1.01 + 1.01^2 + 1.01^3 + \cdots \), the common ratio is \( 1.01 \).

• The common ratio decides whether the series converges:
• If \(|r| < 1\), the series converges.
• If \(|r| \geq 1\), the series diverges.

This careful examination of the common ratio helps refine our approach to analyzing series, ensuring we know how the series will behave as it extends infinitely.

The divergence of a series means that the series does not approach a finite limit as more terms are added. Instead, the sum of its terms increases indefinitely or oscillates, indicating that the series lacks a definitive sum.
For the given geometric series \( 1 + 1.01 + 1.01^2 + 1.01^3 + \cdots \), the common ratio is \( 1.01 \). Since the absolute value of the common ratio \( |1.01| \) is greater than 1, this geometric series diverges. This simply implies that, even if you keep adding more terms, the total will never settle at a single value.

• In general, divergence occurs under these situations:
• If the common ratio \(|r|\) is greater than or equal to 1.
• If the series continually grows without bound or swings wildly.

Understanding whether a series diverges helps in predicting long-term behavior and calculating potential sums.