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The Ratio Test is a handy tool used to determine the convergence of a power series. It works by examining the ratio of consecutive terms in a series. In our exercise, we look at the series \( \sum_{n=1}^{\infty} \frac{x^{n}}{n^{2}} \) and focus on two consecutive terms: \( \frac{x^{n+1}}{(n+1)^2} \) and \( \frac{x^{n}}{n^2} \).

The Ratio Test requires us to find:

• \( \left| \frac{a_{n+1} x^{n+1}}{a_n x^n} \right| \)

Using this formula, we simplify to get \( \left| x \right| \cdot \left( \frac{n^2}{(n+1)^2} \right) \). As \( n \to \infty \), the fraction \( \frac{n^2}{(n+1)^2} \) approaches 1, leaving us with \( \left| x \right| \).

For the series to converge using the Ratio Test, this last expression must be less than 1. Hence,

• \( \left| x \right| < 1 \)

This indicates that the series is convergent within this range.

The radius of convergence is a key aspect of a power series' behavior. It defines an interval around a central value in which the series converges.

In our series \( \sum_{n=1}^{\infty} \frac{x^{n}}{n^{2}} \), the variable \( x \) determines if the series will converge. After applying the Ratio Test, we determined that

• \( \left| x \right| < 1 \)

This gives us a radius of convergence of 1. This means the series converges for values of \( x \) within an open interval from \(-1\) to \(1\). Values within this interval are part of the convergence set. The endpoints of the interval, \( x = 1 \) and \( x = -1 \), need further examination.

This test is used to check the convergence of series where terms alternate in sign. For endpoint evaluation in our power series, let’s consider \( x = -1 \).

The series becomes \( \sum_{n=1}^{\infty} \frac{(-1)^n}{n^2} \), which is an alternating series. It meets the criteria:

• The terms \( \frac{1}{n^2} \) decrease steadily.
• The limit of these terms as \( n \to \infty \) is zero.

Thus, by the Alternating Series Test, the series at \( x = -1 \) converges. Similarly, at \( x = 1 \), the series also converges as \( \sum_{n=1}^{\infty} \frac{1}{n^2} \) is a known convergent series.

Evaluating limits plays a pivotal role in ascertaining the convergence or divergence of a series. It is often used in conjunction with tests like the Ratio Test to define convergence properties.

As shown earlier, the limit:

• \( \lim_{n \to \infty} \left| x \right| \cdot \left( \frac{n^2}{(n+1)^2} \right) \)

simplifies to \( \left| x \right| \) after the ratio within the limit approaches 1.

This crucial step confirms the interval of convergence derived from the Ratio Test. Moreover, the limit ensures that the convergence is consistent across the interval \(-1 < x < 1\), establishing \( \left| x \right| < 1 \) as valid. Thus, our interval of convergence is bounded neatly by evaluating this limit.