Problem 44 - Calculus Early Transcendentals: Pearson New International Edition

Question

Let \(p(n)\) and \(q(n)\) be polynomials in \(n\) with nonnegative coefficients. Give simple conditions that determine the convergence or divergence of \(\sum_{n=1}^{\infty} \frac{p(n)}{q(n)}\).

Step-by-Step Solution

Answer

Converges if \(d_q > d_p\); diverges if \(d_q \leq d_p\).

Step 1: Analyze the Degrees of the Polynomials

The convergence or divergence of the series \(\sum_{n=1}^{\infty} \frac{p(n)}{q(n)}\) depends on the degrees of the polynomials \(p(n)\) and \(q(n)\). If \(d_p\) is the degree of \(p(n)\) and \(d_q\) is the degree of \(q(n)\), the first step is to compare these degrees.

Step 2: Determine the Condition for Convergence

If the degree of \(q(n)\), \(d_q\), is greater than the degree of \(p(n)\), \(d_p\), that is \(d_q > d_p\), the series \(\sum_{n=1}^{\infty} \frac{p(n)}{q(n)}\) converges. This is because the terms \(\frac{p(n)}{q(n)}\) approach zero rapidly as \(n\) becomes large.

Step 3: Determine the Condition for Divergence

If the degree of \(q(n)\), \(d_q\), is less than or equal to the degree of \(p(n)\), \(d_p\), that is \(d_q \leq d_p\), the series \(\sum_{n=1}^{\infty} \frac{p(n)}{q(n)}\) diverges. In these cases, the terms do not decrease to zero sufficiently fast.

Key Concepts

Polynomials

Polynomials are mathematical expressions that consist of variables, coefficients, and exponents. They are typically written in the form: \(p(n) = a_k n^k + a_{k-1} n^{k-1} + \cdots + a_1 n + a_0\), where \(a_k, a_{k-1}, ..., a_1, a_0\) are constant coefficients, and \(n\) is a variable.

• The highest power of the variable \(n\) in the polynomial is called the degree of the polynomial.
• Polynomials can have one or more terms, such as quadratic, cubic, or quartic polynomials.
• They are used in various mathematical and real-world applications, ranging from simple algebraic equations to complex modeling tasks.

Understanding polynomials is essential when analyzing series, as it will help determine how quickly terms in a series approach zero.

Degree Comparison

When dealing with the convergence of a series involving polynomials, one of the crucial steps is degree comparison. The degree of a polynomial is key to understanding how the function behaves for large values of \(n\).

• The degree is the largest exponent of \(n\) that appears in the polynomial.
• To compare two polynomials, such as \(p(n)\) and \(q(n)\), we determine their respective degrees, denoted \(d_p\) and \(d_q\).

In the context of series, determining which polynomial is "stronger" (i.e., grows faster) can dictate the behavior of the series \(\sum_{n=1}^{\infty} \frac{p(n)}{q(n)}\). If \(d_q\) is greater than \(d_p\), the denominator grows faster, aiding in the convergence of the series.

Series Divergence

Series divergence occurs when the sum of the terms in a series does not settle to a fixed value, no matter how large the number of terms.

• For the series \(\sum_{n=1}^{\infty} \frac{p(n)}{q(n)}\), divergence is determined by the relationship between the degrees of \(p(n)\) and \(q(n)\).
• If \(d_p\) is greater than or equal to \(d_q\), the numerator grows fast enough to prevent the terms from approaching zero swiftly.

This means, as \(n\) increases, the fraction \(\frac{p(n)}{q(n)}\) does not decrease rapidly enough, leading to divergence.

Infinite Series

An infinite series is a summation of infinitely many terms. It is usually written as \(\sum_{n=1}^{\infty} a_n\), where \(a_n\) represents the individual terms. The convergence or divergence of this sum is a central theme in calculus and analysis.

• An infinite series converges if its sum approaches a specific finite value as more terms are added.
• Conversely, it diverges if the sum tends to infinity or oscillates without settling on a single value.
• Understanding the behavior of infinite series, especially those involving rational functions like \(\frac{p(n)}{q(n)}\) where \(p(n)\) and \(q(n)\) are polynomials, requires careful analysis of the growth rates of these functions as described with their degrees.

Ultimately, determining convergence or divergence can tell us about the long-term behavior and stability of the processes modeled by such series.