91Ó°ÊÓ

Understanding when a series converges is fundamental in mathematics. Specifically, a series is said to converge if the sum of its infinite terms approaches a finite value as more terms are added. This idea of convergence is pivotal when dealing with infinite series, such as geometric series, that have a specific pattern in how the terms are derived and added.

For a geometric series, where each term is derived by multiplying the previous term by a constant known as the common ratio (denoted as \( r \)), convergence can be neatly determined. In general, a geometric series

• Converges if the absolute value of the common ratio \(|r| < 1\).
• Diverges if \(|r| \geq 1\).

This condition ensures that as we proceed to add more terms, the terms become smaller and in turn, the series approaches a specific sum.

In the exercise mentioned, the series is a geometric one with a common ratio of \((\sqrt{x} - 2)\). Thus, checking if \(|\sqrt{x} - 2| < 1\) helps us verify if and where this series converges.

The interval of convergence is essentially the set of all values for which a given series converges. Finding this interval can sometimes be a straightforward task, especially with geometric series, due to their predictable nature.

In our problem, after establishing that the series converges when \(|\sqrt{x} - 2| < 1\), this inequality gives us a way to explore when the series converges:

• First, solve the inequality \(-1 < \sqrt{x} - 2 < 1\).
• Adding 2 to each part, we get \(1 < \sqrt{x} < 3\).
• Squaring each part to remove the square root, we see that \(1 < x < 9\).

This leads us to the interval of convergence \((1, 9)\), meaning the series converges for all \(x\) between 1 and 9, exclusive.

The ratio test is a powerful tool in determining series convergence, especially useful for series where each term is multiplied by a consistent formula or function. This test helps us assess whether the absolute value of the ratio of consecutive terms points toward a limit that is less than one.

Here's a clearer view of how it works:

• For a series \( \sum a_k \), consider the limit \( \lim_{k \to \infty} \left| \frac{a_{k+1}}{a_k} \right| \).
• If this limit is less than 1, the series converges absolutely.
• If greater than 1, or if it tends toward infinity, the series diverges.
• When the ratio equals 1, the test is inconclusive and other methods are needed.

In our case, with the geometric series involving \((\sqrt{x} - 2)^k\), simplifying the ratio for the consecutive terms brings us to \( \lim_{k \to \infty} |\sqrt{x} - 2| < 1 \). This simplicity often makes the ratio test a favored approach when dealing with power or geometric series, offering a direct path toward identifying the series' convergence behavior.