Problem 1 - Calculus Early Transcendentals: Pearson New International Edition

Question

\(\sum_{k=0}^{\infty} \frac{1}{k+3}\)

Step-by-Step Solution

Answer

The series \( \sum_{k=0}^{\infty} \frac{1}{k+3} \) diverges.

Step 1: Understand the Series

The series given is the sum of terms \(\frac{1}{k+3}\) starting from \(k = 0\) and going to infinity. This is an infinite series where each term is of the form \(\frac{1}{n}\), which resembles a \(p\)-series with \(p = 1\).

Step 2: Determine the Convergence

A \(p\)-series of the form \sum_{n=1}^{\infty} \frac{1}{n^p}\ converges only if \(p > 1\). Our series can be compared to \(\sum_{n=3}^{\infty} \frac{1}{n}\), which has \(p = 1\) and is known as the harmonic series, which diverges.

Step 3: Apply the Integral Test

To confirm divergence, use the integral test, which states that if \(f(n)\) is a positive, continuous, and decreasing function for \(n \geq 3\), then \(\int_{3}^{\infty} \frac{1}{x} \, dx\) can be used to test convergence. Calculate \(\int_{3}^{\infty} \frac{1}{x} \, dx = \lim_{b \to \infty} \left( \ln|x| \right)_{3}^{b} = \lim_{b \to \infty} [\ln b - \ln 3] = \infty\), confirming divergence.

Key Concepts

p-series

A p-series is a type of infinite series that features prominently in mathematical analysis. The general form of a p-series is \[ \sum_{n=1}^{\infty} \frac{1}{n^p} \]where \( p \) is a positive real number. One of the most crucial aspects of p-series is determining their convergence. The convergence of a p-series depends on the value of \( p \). Here’s how it works:

• If \( p > 1 \), the series converges. This means that the sum of the infinite number of terms results in a finite value.
• If \( p \leq 1 \), the series diverges, indicating that the sum of the series tends towards infinity.

In the provided exercise, the series \( \sum_{k=0}^{\infty} \frac{1}{k+3} \) can be related to a p-series with \( p = 1 \). Thus, according to the rule outlined, this series does not converge.

harmonic series

The harmonic series is a well-known and classical example of a p-series with \( p = 1 \). It takes the form:\[\sum_{n=1}^{\infty} \frac{1}{n}\]The harmonic series is fascinating because despite its terms getting smaller as \( n \) increases, it diverges. This means that if you keep adding terms from the series, the sum will eventually become infinitely large.

A useful comparison is with a series like \( \sum_{k=0}^{\infty} \frac{1}{k+3} \) from the exercise, which starts its sequence at \( n = 3 \). This shifted version still behaves like a harmonic series and thus also diverges. Understanding the harmonic series supports recognizing series that appear similar in their divergence patterns.

integral test

The integral test is a powerful tool for assessing the convergence of infinite series. When you have a series \( \sum_{n=1}^{\infty} a_n \), it can be useful when \( a_n = f(n) \) for some function \( f \), and \( f(x) \) is:

• Positive
• Continuous
• Decreasing for \( x \geq a \)

To apply the integral test, you calculate the improper integral:\[\int_{a}^{\infty} f(x) \, dx\]If the integral converges, so does the series. Conversely, if the integral diverges, the series diverges as well.

In our exercise, \( f(x) = \frac{1}{x} \) starting from \( x = 3 \) aligns perfectly to use the integral test. Evaluating \[\int_{3}^{\infty} \frac{1}{x} \, dx \]yields \( \infty \), confirming divergence.

divergence of series

A series is said to diverge if its sequence of partial sums does not approach a finite limit. Instead, these sums continue to grow indefinitely. Divergence might not always be apparent on first looks, especially with series that have terms decreasing in size.

There are several ways to determine if a series diverges, including:

• Comparison test against known divergent series like the harmonic series.
• Application of the integral test, as shown in the exercise.
• Checking if the terms don’t approach zero, a necessary condition for convergence.

The exercise piece \( \sum_{k=0}^{\infty} \frac{1}{k+3} \) demonstrates this concept well. Although each term \( \frac{1}{k+3} \) seems to shrink, the entire series still diverges, much like the harmonic series its behavior mimics. Understanding divergence is essential for determining the behavior of infinite series.