Problem 14 - Essential Calculus Early Transcendentals

Question

Determine whether the series is convergent or divergent. If it is convergent, find its sum. $$\sum_{k=1}^{\infty} \frac{k(k+2)}{(k+3)^{2}}$$

Step-by-Step Solution

Answer

The series is divergent by the limit comparison test.

Step 1: Identify the Type of Series

The series given is \( \sum_{k=1}^{\infty} \frac{k(k+2)}{(k+3)^{2}} \). It is an infinite series with terms that involve a rational function of \( k \).

Step 2: Simplify the Expression

Simplify the expression \( \frac{k(k+2)}{(k+3)^{2}} \). The degree of the numerator (which is 2) is equal to the degree of the denominator (also 2). This hints that the series might behave like a divergent series, such as \( \sum \frac{1}{k} \).

Step 3: Apply the Limit Comparison Test

Select a comparable series for which we know the behavior. We'll use \( b_k = \frac{k^2}{k^2} = 1 \). Apply the limit comparison test:\[L = \lim_{k \to \infty} \frac{\frac{k(k+2)}{(k+3)^{2}}}{\frac{1}{k}} = \lim_{k \to \infty} \left( \frac{k(k+2)k}{(k+3)^2} \right) = \lim_{k \to \infty} \frac{k^3 + 2k^2}{k^3 + 6k^2 + 9k}.\]Simplifying gives:\[L = \lim_{k \to \infty} \frac{1 + \frac{2}{k}}{1 + \frac{6}{k} + \frac{9}{k^2}} = 1.\]

Step 4: Determine Convergence or Divergence

Since \( L = 1 \) and the series \( \sum \frac{1}{k} \) is divergent, according to the limit comparison test, the series \( \sum_{k=1}^{\infty} \frac{k(k+2)}{(k+3)^{2}} \) also diverges.

Key Concepts

Convergence and Divergence

Convergence and divergence are key concepts when dealing with infinite series. In simple terms, a series converges if the sum of its terms approaches a finite number as the number of terms goes to infinity. On the other hand, a series diverges if the sum grows indefinitely, or doesn't settle at a fixed value.

To determine whether a series is convergent or divergent, various tests can be used. One common method is the Limit Comparison Test. This test compares the given series to a benchmark series whose behavior is already known. If the series behaves similarly, then its convergence or divergence will be the same. This is particularly useful for rational function series, where terms are fractions involving polynomial expressions.

Understanding whether an infinite series converges or diverges is crucial in mathematics and physics, as it impacts how functions behave at infinity and can influence outcomes in real-world applications.

Infinite Series

An infinite series is a sum of infinitely many terms. Formulated as \( \sum_{k=1}^{\infty} a_k \), it involves adding up terms one after another, reaching towards infinity. Some series will "settle" at a number (converge) while others either grow endlessly or fail to fix at a specific value (diverge).

There are many kinds of infinite series, such as geometric, harmonic, and the series of rational functions. Each type can exhibit unique properties and require different methods for testing convergence. For instance, geometric series can be easily tested for convergence using their common ratio, while more complex rational series might need special tests like the Limit Comparison Test.

Infinite series are significant because they can model various phenomena, from simple patterns in nature to complex structures in engineering and science. They also help in understanding the behavior of functions that can be expressed as series, such as Taylor and Fourier series.

Rational Function Series

A rational function series involves terms that are ratios of polynomials. In the provided series, \( \sum_{k=1}^{\infty} \frac{k(k+2)}{(k+3)^2} \), each term is a fraction formed by polynomials of \( k \). Understanding the degree of the numerator and the denominator is crucial here as it helps predict the behavior of the series.

If the degree of the numerator exceeds that of the denominator, the series generally behaves divergent, similar to a series like \( \sum \frac{1}{k} \). However, when these degrees are equal, the series might still diverge, specifically mimicking the behavior of harmonic series, which are known to diverge. This is because the terms don't decay fast enough to sum up to a finite limit.

Rational function series are often studied using the Limit Comparison Test to draw parallels with simpler series. Practically, these series appear in various fields, such as physics and engineering, where the physical systems can be modeled through rational functions.