91Ó°ÊÓ

In calculus, when dealing with series, a key concept is the **radius of convergence**. This helps to determine how far a power series converges around a certain point. For the series given in the exercise, we use the **Ratio Test** to find this radius. The **Ratio Test** involves taking the limit of the absolute value of the ratio between consecutive terms of the series. For our series, we derive:

• First, determine the terms: \(a_n = \frac{(x-1)^n}{\sqrt{n}}\).
• Then, it calculates to: \( \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = \left|x-1\right| \cdot \lim_{n \to \infty} \sqrt{\frac{n}{n+1}} = |x-1| \).

Since this must be less than 1 for convergence, we find \(|x-1| < 1\), giving us the radius of convergence \(R = 1\). This tells us that the series will converge when the variable \(x\) is within 1 unit from 1, as discussed in the next section on the interval of convergence.

Determining the **interval of convergence** involves taking the radius from the radius of convergence and applying it to the series. Since we found that \(|x-1| < 1\), it follows that the center of the interval is 1. Applying the radius, our interval starts at 0 and ends at 2:

• Subtract the radius from the center: \(1 - 1 = 0\)
• Add the radius to the center: \(1 + 1 = 2\)

Thus, the interval of convergence is \( (0, 2) \). It's crucial to check the endpoints separately:

• At \(x = 0\), the series changes, and needs evaluation using tests for convergence.
• Similarly, at \(x = 2\), we need to reassess the series' behavior.

This method ensures that we understand not just how far the series converges, but precisely where it does so including edge points.

**Absolute convergence** refers to a series converging even when we consider the absolute value of its terms. When finding absolute convergence, we test whether the series:1. Converges completely on the interval found previously.2. Holds for every individual value within this interval, including possible endpoints.For our particular series:- It converges absolutely for all values strictly between the bounds of our interval, i.e., **\(0 < x < 2\)**.Checking for absolute convergence involves verifying that the sum of the absolute values of the series converges. Within this interval, the series does meet this criteria, thus showing its consistent behavior through the interval except possibly at boundaries, which must be evaluated separately.

**Conditional convergence** occurs when a series converges in its entirety but fails to converge when we consider absolute values. Essentially, this means the series converges but isn't absolutely convergent.

The series in question shows:

• At \(x = 0\), the series simplifies to an alternating series \(\sum_{n=1}^{\infty} \frac{(-1)^n}{\sqrt{n}}\). The **Alternating Series Test** helps confirm that this converges conditionally since the terms decrease in absolute value and approach zero.
• At \(x = 2\), it diverges entirely as it becomes a p-series with a negative exponent greater than -1.

Thus, conditional convergence is specifically identifiable only when checking individual points like these endpoints within our interval.