91Ó°ÊÓ

When examining the convergence of an infinite series, one effective approach is to use the p-series test. A p-series is an infinite series of the form \( \sum_{n=1}^{\infty} \frac{1}{n^p} \), where \( p \) is a constant. The convergence or divergence of a p-series depends on the value of \( p \).

According to the p-series test, a p-series is convergent if \( p > 1 \) and divergent if \( p \leq 1 \). This is because when \( p > 1 \) the terms of the series decrease quickly enough that their sum approaches a finite limit. Conversely, when \( p \leq 1 \), the terms do not decrease fast enough and the series 'grows' infinitely.

In the context of the exercise, the series \( \sum_{n=1}^{\infty} \frac{3}{n \sqrt{n}} \) can be simplified to \( 3n^{-1.5} \) which resembles the form of a p-series with \( p = 1.5 \), greater than 1. Hence, applying the p-series test immediately suggests its convergence without the need for further complex calculations.

An infinite series is essentially the sum of an infinite sequence of numbers. It can be expressed in the form \( \sum_{n=1}^{\infty} a_n \), where \( a_n \) represents the nth term of the sequence. The intricate nature of infinite series requires specific methods for determining whether the sum is finite (convergent) or if it increases without bound (divergent).

The concept of infinite series is central to many areas of mathematics, particularly analysis and calculus, because it allows us to describe functions and number sets in compact forms. Furthermore, infinite series can model complex phenomena in physics and other sciences where finite sums are not sufficient.

Recognizing the form an infinite series takes, such as whether it is arithmetic, geometric, or a p-series, provides the necessary first step towards determining its convergence, as can be seen in the given exercise through the identification of the series as a p-series.

A series is said to be convergent if the sum of its infinite terms approaches a specific value as n approaches infinity. This means that there is a clear limit to the sum despite the series being infinite. To determine if a series is convergent, various tests such as the p-series test, comparison test, ratio test, and others may be applied.

In practice, establishing convergence is critical for confirming that processes described by the series are well-defined and finite in nature. For example, in the field of series solutions to differential equations, convergence ensures that the obtained power series solution is meaningful and can represent a function over some interval.

Reflecting on the exercise provided, since the series \( \sum_{n=1}^{\infty} \frac{3}{n \sqrt{n}} \) is identified as a convergent p-series, it reassures us that the series has a finite limit, which is an integral conclusion in many practical applications.