Problem 28 - Thomas Calculus

Question

Which of the series in Exercises \(11-40\) converge, and which diverge? Give reasons for your answers. (When you check an answer, remember that there may be more than one way to determine the series' convergence or divergence.) $$ \sum_{n=1}^{\infty} \frac{1}{\sqrt{n}(\sqrt{n}+1)} $$

Step-by-Step Solution

Answer

The series \( \sum_{n=1}^{\infty} \frac{1}{\sqrt{n}(\sqrt{n}+1)} \) diverges due to its similarity to the divergent harmonic series.

Step 1: Simplify the Series

Let's start by simplifying the general term of the series, \( \frac{1}{\sqrt{n}(\sqrt{n}+1)} \), to make it easier to analyze. Notice that we can factor the denominator: \( \sqrt{n}(\sqrt{n}+1) = n + \sqrt{n} \). This simplified form might help us decide the behavior of the series.

Step 2: Determine the Dominant Term

The term \( n + \sqrt{n} \) in the denominator means \( \frac{1}{\sqrt{n}(\sqrt{n}+1)} = \frac{1}{n + \sqrt{n}} \approx \frac{1}{n} \) as \( n \to \infty \). This suggests that the series resembles the harmonic series.

Step 3: Compare with the Harmonic Series

The harmonic series \( \sum_{n=1}^{\infty} \frac{1}{n} \) is a well-known divergent series. Since our original series \( \sum_{n=1}^{\infty} \frac{1}{n + \sqrt{n}} \) resembles \( \sum_{n=1}^{\infty} \frac{1}{n} \) when \( n \) is large, we can use the Limit Comparison Test to rigorously compare them.

Step 4: Apply the Limit Comparison Test

To use the Limit Comparison Test, compute \( \lim_{n \to \infty} \frac{\frac{1}{\sqrt{n}(\sqrt{n}+1)}}{\frac{1}{n}} = \lim_{n \to \infty} \frac{n}{n + \sqrt{n}} \). Simplifying, this limit becomes \( \lim_{n \to \infty} \frac{1}{1 + \frac{1}{\sqrt{n}}} = 1 \). Since this limit is a positive constant, both series will converge or diverge together.

Step 5: Conclusion of Convergence/Divergence

Our comparison found that the test limit is 1, a positive constant. Hence, \( \sum_{n=1}^{\infty} \frac{1}{\sqrt{n}(\sqrt{n}+1)} \) diverges because it behaves similarly to the harmonic series, which is divergent.

Key Concepts

Limit Comparison Test

The Limit Comparison Test is a powerful tool when we need to determine the convergence or divergence of a series whose terms are not immediately comparable to a known series. To apply the test, we compare the given series to another series with known behavior, like the harmonic series. Here's how it works:

• Identify another series that resembles the behavior of your original series as terms approach infinity.
• Calculate the limit, \( \lim_{n \to \infty} \frac{a_n}{b_n} \), where \( a_n \) and \( b_n \) are terms from the original and known series, respectively.
• If the limit is a positive finite value, both series converge or diverge together.

In the exercise, our series is compared to the harmonic series, which is divergent, through this method. The limit was found to be 1, indicating that both series exhibit the same divergent behavior.

Harmonic Series

The harmonic series is a classic example in mathematics, often cited in textbooks for its simple form yet intriguingly divergent nature. It is expressed as: \[\sum_{n=1}^{\infty} \frac{1}{n}\]

This series appears to converge, as the terms \( \frac{1}{n} \) become very small as \( n \) increases. However, appearances can be deceiving, and the harmonic series is a famous example of a divergent series:

• The sum of its terms grows indefinitely, albeit slowly, as more terms are added.
• This is a surprise because each added term becomes smaller, yet their sum never stabilizes.

Understanding the divergence of the harmonic series helps in analyzing other similar series, like the one in the exercise.

Divergent Series

A divergent series is one that does not have a finite sum. Unlike convergent series, whose terms effectively "collapse" into a finite value, divergent series continue to grow as more terms are added. Sometimes, they grow so slowly that they appear almost stable, but ultimately, they lack a limit.

In relation to the exercise, divergence can be observed by noting that if a series has a dominant term similar to a known divergent series, the suspected series may also diverge when other terms do not sufficiently curb its growth. This insight is foundational in using the Limit Comparison Test, as demonstrated when our series was likened to the harmonic series, a key example of divergence in action.

Simplification of Series

Simplifying complex series expressions is often the first step in determining their behavior. By transforming the series into a more familiar form, we can better understand its properties relative to convergence or divergence.

• Simplification can involve algebraic manipulation, such as factoring or approximating complex terms.
• A clearer picture emerges when a series is reduced, often allowing us to compare it more easily to a benchmark series like the harmonic series.
• In the problem at hand, breaking down the series into \( \frac{1}{n + \sqrt{n}} \) highlighted its resemblance to the harmonic series.

This simplification guided the subsequent steps in applying the Limit Comparison Test, confirming the series' divergence.