Problem 29 - Thomas Calculus

Question

Determining Convergence or Divergence In Exercises \(17-44,\) use any method to determine if the series converges or diverges. Give reasons for your answer. $$\sum_{n=1}^{\infty}\left(\frac{1}{n}-\frac{1}{n^{2}}\right)$$

Step-by-Step Solution

Answer

The series diverges by the Limit Comparison Test.

Step 1: Simplify the series

Start by rewriting the expression for the series term: \( \frac{1}{n} - \frac{1}{n^2} = \frac{n-1}{n^2} \). Thus, the series becomes \( \sum_{n=1}^{\infty} \frac{n-1}{n^2} \).

Step 2: Compare with Harmonic Series

Observe that \( \frac{n-1}{n^2} \approx \frac{1}{n} \) for large values of \( n \). The series \( \sum_{n=1}^{\infty} \frac{1}{n} \) is a well-known harmonic series, which diverges.

Step 3: Use Limit Comparison Test

To further analyze, apply the Limit Comparison Test. Let \( a_n = \frac{n-1}{n^2} \) and \( b_n = \frac{1}{n} \). Compute the limit \( \lim_{n \to \infty} \frac{a_n}{b_n} = \lim_{n \to \infty} \frac{n-1}{n^2} \cdot n = \lim_{n \to \infty} \frac{n-1}{n} = 1 \).

Step 4: Conclusion from Limit Comparison Test

Since the limit in Step 3 is a positive finite number (1), and \( \sum_{n=1}^{\infty} \frac{1}{n} \) diverges, by the Limit Comparison Test, the original series \( \sum_{n=1}^{\infty} \frac{n-1}{n^2} \) also diverges.

Key Concepts

Limit Comparison Test

The Limit Comparison Test is a valuable tool for determining the convergence or divergence of a series by comparing it to another series whose behavior is already known. To use this test, you'll need two series with positive terms: the one you are investigating, denoted as \( a_n \), and a simpler comparison series, \( b_n \).
Here's how the process works:

• Calculate the limit \( \lim_{n \to \infty} \frac{a_n}{b_n} \).
• If the limit is a positive finite number (meaning it is greater than 0 and less than infinity), then both series either converge or diverge together.

For example, if you're working with the series \( \sum_{n=1}^{\infty} \frac{n-1}{n^2} \), you might compare it to \( \sum_{n=1}^{\infty} \frac{1}{n} \), a known harmonic series that diverges. This comparison helps simplify your understanding of the original series by leveraging a known outcome.

Harmonic Series

The Harmonic Series is one of the first series that students learn about in the context of convergence and divergence. It is given by \( \sum_{n=1}^{\infty} \frac{1}{n} \).

Why is the Harmonic Series important? Because it is a classic example of a diverging series, meaning it does not settle at a finite sum. No matter how many terms you add up, the total just keeps getting larger without bound.

Here is a simplified explanation:

• Each term of the Harmonic Series gets smaller, yet not fast enough to make the series as a whole converge.
• This behavior makes it a perfect candidate for comparison when using tests like the Limit Comparison Test.

Its divergence is a useful "benchmark" or reference point when assessing other series for convergence.

Divergence and Convergence

Determining whether a series diverges or converges is crucial in understanding its sum behavior across an infinite number of terms. Convergence implies that the series adds up to a specific, finite value. In contrast, divergence means the sum either grows indefinitely or fails to settle to a single value.

There are several key points to remember:

• If the series converges, each sequence of its partial sums approaches a specific value as you include more terms.
• In contrast, a divergent series won't bring its partial sums to any particular limit as you include more terms.
• Divergence can occur even if the terms of the series get smaller—illustrated by the Harmonic Series.

Utilizing tests like the Limit Comparison Test helps to assess whether a series converges or diverges, making these concepts foundational in the study of infinite series.