Problem 32 - Thomas Calculus

Question

In Exercises \(31-38,\) use the \(n\) th-Term Test for divergence to show that the series is divergent, or state that the test is inconclusive. $$\sum_{n=1}^{\infty} \frac{n(n+1)}{(n+2)(n+3)}$$

Step-by-Step Solution

Answer

The series diverges as the n-th term approaches 1, not zero.

Step 1: Identify the Series

We are given the series \( \sum_{n=1}^{\infty} \frac{n(n+1)}{(n+2)(n+3)} \). Our task is to determine if this series is divergent using the \(n\)th-Term Test for divergence.

Step 2: Apply the n-th Term Divergence Test

The \(n\)th-Term Test states that if the limit of the terms of the series as \(n\) approaches infinity is not zero, then the series diverges. We start by finding the limit of \( \frac{n(n+1)}{(n+2)(n+3)} \) as \(n\) approaches infinity.

Step 3: Simplify the Expression for the Limit

To find \( \lim_{n \to \infty} \frac{n(n+1)}{(n+2)(n+3)} \), notice that the highest power of \(n\) in the numerator is \(n^2\) and in the denominator is also \(n^2\). Divide every term by \(n^2\) to obtain: \[\lim_{n \to \infty} \frac{n(n+1)}{(n+2)(n+3)} = \lim_{n \to \infty} \frac{1 + \frac{1}{n}}{1 + \frac{5}{n} + \frac{6}{n^2}} = \lim_{n \to \infty} \frac{1 + \frac{1}{n}}{1 + \frac{5}{n}}.\]

Step 4: Evaluate the Limit

As \(n\) approaches infinity, \( \frac{1}{n} \) and \( \frac{5}{n} \) both approach zero. Thus, the expression simplifies to:\[\lim_{n \to \infty} \frac{1 + \frac{1}{n}}{1 + \frac{5}{n}} = \frac{1 + 0}{1 + 0} = 1.\]Since this limit is not zero, the \(n\)th-Term Test for divergence tells us that this series diverges.

Step 5: Conclusion

Based on the \(n\)th-Term it was determined that since the limit of the terms is not zero, the series \( \sum_{n=1}^{\infty} \frac{n(n+1)}{(n+2)(n+3)} \) is divergent.

Key Concepts

Divergent Series

When we talk about divergent series, we refer to a series that does not converge to a specific finite value. A series is an endless sum of numbers. For example, a series like \( \sum_{n=1}^{\infty} a_n \) is divergent if, as you add up more and more terms, the total sum keeps growing without bounds or does not settle close to any particular number.
Understanding if a series is divergent is important because it tells us about the behavior of the infinite sum. Divergent series might seem tricky, but they can be identified using various tests like the n-th Term Test. If you identify a series as divergent, it means that the sum does not settle down to a single number. More formally, if the partial sums of the series do not approach a finite limit, the series is divergent.
Recognizing divergent series helps in analyzing mathematical models and real-life applications. It guides us on when series can be practically used, like in calculations and predictions.

Limit of a Sequence

The concept of the limit of a sequence is fundamental in understanding series. A sequence is simply a list of numbers in a specific order, such as \( a_1, a_2, a_3, \ldots \). As we study these numbers, the limit identifies the number that the sequence's terms approach as they progress to infinity. When we talk about limits, we use notation like \( \lim_{n \to \infty} a_n \).
For instance, if adding more and more terms of the sequence results in getting closer and closer to a particular value, that value is the limit of the sequence. This concept is crucial because the behavior of a series often hinges on the limit of its sequence of terms. For a series to converge, the terms of the sequence must approach zero, while if the limit is not zero, the series might be divergent.
Understanding limits helps us evaluate the potential outcomes of mathematical problems. It goes beyond theory and becomes a useful tool in fields like calculus, where behavior near infinity is examined.

Rational Functions

Rational functions are a type of function expressed as the ratio of two polynomials. These functions look like \( \frac{P(x)}{Q(x)} \), where both \(P(x)\) and \(Q(x)\) are polynomials. The degree of the polynomial in the numerator might be the same as, larger than, or smaller than the degree of the polynomial in the denominator.
In the context of sequences and series, rational functions often determine the terms we need to analyze. For the given series \( \sum_{n=1}^{\infty} \frac{n(n+1)}{(n+2)(n+3)} \), each term is a rational function. We examine these functions to understand how they behave as \( n \) becomes very large.
Rational functions often simplify to show their tendencies as their variable approaches infinity. This simplification helps in applying tests like the n-th Term Test for divergence. Recognizing that these functions have comparable powers in both the numerator and denominator aids in assessing the series behavior over the long haul.

Convergence Tests

Convergence tests are essential in determining whether a series converges or diverges. These tests are built to apply specific criteria to check the behavior of series or sequence terms. One such test is the n-th Term Test for Divergence, which is quite straightforward but very useful.

• **n-th Term Test:** It states if the limit of the sequence’s terms is not zero, the series diverges. This test alone cannot confirm convergence, but divergence can be identified easily.

Understanding these tests helps us classify series and designate their ends. These tests include criteria like the ratio test, root test, and the aforementioned n-th Term Test, among others. They help us look beyond individual terms of the series and find out how the entire sum behaves.
Employing convergence tests is like solving a puzzle. Each piece, or test, gives insight into the series, helping determine if it brings value by converging, or if it eludes definition by diverging. Knowing these tests is crucial, especially for work involving infinite series, to make informed predictions about their behavior.