91Ó°ÊÓ

When examining the convergence of a series, identifying the dominant term is crucial. The dominant term in a series is the term that influences the behavior of the series as the variable, often denoted as \( n \), approaches infinity. In the series given by \( \sum \frac{5-2\sqrt{n}}{n^3} \), the term \( a_n = \frac{5}{n^3} - \frac{2}{n^{2.5}} \) can be split into two parts: \( \frac{5}{n^3} \) and \( -\frac{2}{n^{2.5}} \).

As \( n \) grows larger, both terms trend towards zero. However, it's essential to determine which term diminishes slower or retains its influence longer. Between the two, \( \frac{2}{n^{2.5}} \) decreases at a slower rate compared to \( \frac{5}{n^3} \) because its exponent in the denominator, 2.5, is smaller than 3. This makes \( \frac{2}{n^{2.5}} \) the dominant term, meaning its convergence properties will determine those of the series. By focusing on this term, we simplify complex series into a more manageable form to analyze convergence.

The Comparison Test is a valuable tool for understanding series convergence. It compares a given series to a reference series that is already known to be convergent or divergent. Here’s how it works: if the terms \( a_n \) of a series are less than or equal to the terms \( b_n \) of a known convergent series, then the series with terms \( a_n \) is also convergent. Conversely, if \( a_n \) are greater than or equal to the terms \( b_n \) of a known divergent series, then the series with terms \( a_n \) is also divergent.

In our specific case, we've identified \( \frac{2}{n^{2.5}} \) as the dominant term of the series. By comparing it with \( \sum \frac{1}{n^{2.5}} \), which is a known p-series, we can determine convergence. As the series \( \sum \frac{1}{n^{2.5}} \) converges (since \( p = 2.5 > 1 \)), our series \( \sum \frac{5-2\sqrt{n}}{n^3} \) also converges when compared correctly using the context provided by the Comparison Test principles.

Understanding p-series is foundational in analysis, especially in series convergence. A p-series is an infinite series of the form \( \sum \frac{1}{n^p} \), where \( p \) is a positive real number. The convergence behavior of a p-series is straightforward:

• If \( p > 1 \), the series converges.
• If \( p \leq 1 \), the series diverges.

In our exercise, the dominant term, \( \frac{2}{n^{2.5}} \), is akin to a p-series with \( p = 2.5 \). Since 2.5 is greater than 1, it ensures convergence. Recognizing this connection provides a powerful shortcut in series analysis. Instead of testing complex each term, we rely on the established rules for p-series.

Thus, the question of whether the series \( \sum \frac{5-2\sqrt{n}}{n^3} \) converges is answered by comparing its similarity to a relatively straightforward p-series. Its net behavior is governed by the p-series property, streamlining analysis and encouraging a deeper mathematical understanding.