Problem 56 - Essential Calculus

Question

Determine the convergence or divergence of the series. $$ \sum_{n=1}^{\infty} \frac{1}{n(n+3)} $$

Step-by-Step Solution

Answer

The provided series diverges, as shown by the limit comparison test with the comparison sequence \(\frac{1}{n}\).

Step 1: Identify the comparison sequence

An obvious choice for a comparison sequence in the limit comparison test is \(\frac{1}{n}\), as the behavior of the harmonic series is well known. It's a simple sequence that allows easy simplification and is known to diverge. The harmonic series \( \sum_{n=1}^\infty \frac{1}{n} \) is a standard p-series that diverges, a basis for comparison.

Step 2: Perform the limit comparison

Next, calculate the limit of the ratio as \(n\) approaches infinity: \[ \lim_{n\to\infty}\frac{\frac{1}{n(n+3)}}{\frac{1}{n}} = \lim_{n\to\infty}\frac{n}{n+3} = 1 \]

Step 3: Interpret the result

Since the limit is a finite, positive number, the given sequence and the comparison sequence \(\frac{1}{n}\) share the same fate. The comparison sequence, which is known to diverge, thus indicates that the given series also diverges.

Key Concepts

Limit Comparison Test

The Limit Comparison Test is a valuable tool in determining the convergence or divergence of series. This test comes into play when direct comparison isn't straightforward or when comparing with known series is advantageous.

To use the Limit Comparison Test, you follow these steps:

• Identify two series: the series you're investigating and a comparison series whose behavior (convergence or divergence) is known.
• Take the limit of the ratio of the terms of these two series as the index approaches infinity. For series \( \sum a_n \) and \( \sum b_n \), calculate \( \lim_{n\to\infty}\frac{a_n}{b_n} \).
• If the limit is a positive finite number, both series either converge or diverge. The known behavior of \( \sum b_n \) gives you the answer for \( \sum a_n \).

In our example, the limit comparison with the harmonic series \( \sum \frac{1}{n} \), which diverges, showed that the series \( \sum \frac{1}{n(n+3)} \) also diverges.

Harmonic Series

The harmonic series is one of the most famous examples of a divergent series. It is given by \( \sum_{n=1}^{\infty} \frac{1}{n} \). Even though the terms of this series decrease to zero, the entire series diverges, which can be surprising.

This series is important for a few reasons:

• Its divergence highlights a key point: series with terms that go to zero don't necessarily converge.
• The series serves as an essential benchmark for comparisons, as seen in many tests.

In our exercise, the harmonic series is used for comparison in the Limit Comparison Test to show that the series \( \sum \frac{1}{n(n+3)} \) shares its divergent nature.

P-Series Divergence

P-series are types of series denoted by \( \sum_{n=1}^{\infty} \frac{1}{n^p} \), where \( p \) is a real number. The behavior of a p-series depends on the value of \( p \):

• The series converges if \( p > 1 \).
• The series diverges if \( p \leq 1 \).

This classification makes p-series handy for quickly assessing the behavior of similar looking series. For instance, the harmonic series \( \sum \frac{1}{n} \) is a p-series where \( p = 1 \), hence it diverges.

By understanding the nature of the p-series, students can utilize it not only as a comparison, like in the Limit Comparison Test but also as a foundational concept to explore other types of series.