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The radius of convergence of a power series is a vital concept in determining where a series converges. It measures the distance from the center of the series within which the series converges. For any power series \( \sum_{n=0}^{\infty} a_n (x - c)^n \), the series converges when \( |x - c| < R \) and diverges when \( |x - c| > R \), where \( R \) is the radius of convergence.
To find this radius, we often use the Ratio Test to calculate the limit, which simplifies to \( |x| < R \). For the given series, following the calculation steps yields \( R = 1 \). This means the series converges for all \( x \) values within a distance of 1 from the center, typically at \( x = 0 \) unless shifted.

The interval of convergence is the specific range of x values for which the series actually converges. It is determined once the radius of convergence is known. However, special attention is required to test the endpoints of the interval: \( x = c - R \) and \( x = c + R \).
After finding \( R = 1 \) as the radius of convergence for our series, the potential interval is \( -1 < x < 1 \). However, to determine whether the endpoints belong to the interval, we must test them separately.

• For \( x = 1 \): The series diverges.
• For \( x = -1 \): The series converges.

Thus, the interval of convergence is \([-1, 1)\). This interval indicates that at \( x = -1 \), the series converges, but at \( x = 1 \), it does not.

The Ratio Test is a helpful tool in determining convergence of a series by examining the ratio of successive terms. Specifically, it looks at the limit \( \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| \).
If the limit \( L < 1 \), the series converges absolutely. If \( L > 1 \), or the limit is infinite, the series diverges. If \( L = 1 \), the test is inconclusive.
In our problem, applying the Ratio Test involved computing \( \lim_{n \to \infty} \left| \frac{n}{n+1} \right| \cdot |x| = |x| \), which simplifies to \( |x| < 1 \) for convergence. Hence, the ratio test helped find our radius of convergence \( R = 1 \).

The Alternating Series Test is useful when dealing with series whose terms alternate in sign, such as \( \sum (-1)^n a_n \). For convergence, two conditions must be met:

• The absolute value of the terms \( a_n \) must be decreasing, i.e., \( a_{n+1} \le a_n \).
• The sequence \( a_n \) should tend to zero as \( n \) approaches infinity.

In our series, at endpoint \( x = -1 \), the terms are of the form \( \frac{(-1)^n}{2n-1} \), which clearly alternate. Here, \( a_n = \frac{1}{2n-1} \) is indeed decreasing and tends to zero. Thus, the Alternating Series Test confirms convergence at \( x = -1 \). This highlights why the endpoint \( x = -1 \) is included in the interval of convergence.