Question

Use the Limit Comparison Test to determine the convergence of the given series; state what series is used for comparison. $$\sum_{n=1}^{\infty} \frac{1}{n+\sqrt{n}}$$

Short answer

The series diverges; compared with the harmonic series.

Step-by-step solution

Choose a Comparison Series

To use the Limit Comparison Test, we need a simpler series to compare with. Observe that in the expression \( \frac{1}{n + \sqrt{n}} \), the term \( n \) dominates \( \sqrt{n} \) for large \( n \). Hence, a natural choice for comparison is \( \sum_{n=1}^{\infty} \frac{1}{n} \), which is the harmonic series.

Apply the Limit Comparison Test

For the Limit Comparison Test, compute \( \lim_{n \to \infty} \frac{a_n}{b_n} \), where \( a_n = \frac{1}{n + \sqrt{n}} \) and \( b_n = \frac{1}{n} \), to determine the relationship between the series. This limit becomes \( \lim_{n \to \infty} \frac{n}{n + \sqrt{n}} = \lim_{n \to \infty} \frac{1}{1 + \frac{1}{\sqrt{n}}} \).

Evaluate the Limit

As \( n \to \infty \), the term \( \frac{1}{\sqrt{n}} \to 0 \). Therefore, the limit \( \frac{1}{1 + \frac{1}{\sqrt{n}}} \to 1\). Since the limit is a positive finite number \( c = 1 \), by the Limit Comparison Test, both series \( \sum \frac{1}{n + \sqrt{n}} \) and \( \sum \frac{1}{n} \) either converge or diverge together.

Conclusion about Convergence

The harmonic series \( \sum \frac{1}{n} \) is known to diverge. By the Limit Comparison Test, since the limit was positive and finite, \( \sum \frac{1}{n + \sqrt{n}} \) must also diverge.

Key concepts

Convergence of Series

Convergence is an important concept in the analysis of series in mathematics. A series is said to converge when the sum of its terms approaches a specific number as the number of terms increases. Understanding whether a series converges or not helps determine if there is a specific sum that the series approaches.
In the context of the Limit Comparison Test, convergence is assessed by comparing a complex series with a more familiar one. If the comparison reveals that both series behave similarly (either both converge or both diverge), it helps us make conclusions about the original series. The Limit Comparison Test uses a limit to evaluate the relationship between series terms. This limit tells us whether the series we're studying has a similar behavior to the chosen comparison series and helps us decide about its convergence.

Harmonic Series

The harmonic series is a fundamental sequence in mathematics. It is typically expressed as \( \sum_{n=1}^{\infty} \frac{1}{n} \). Each term in the harmonic series is the reciprocal of an integer. Although it might seem like the harmonic series should converge because its terms get smaller, it actually diverges.
The divergence of the harmonic series is a classic discovery in calculus. As you continue adding more and more terms, the sum increases without bound. This property makes the harmonic series a useful tool for comparison in convergence tests, such as the Limit Comparison Test.
When analyzing series like \( \sum \frac{1}{n + \sqrt{n}} \), the harmonic series can simplify the process of determining convergence or divergence. By comparing the terms with the harmonic series, one can leverage its divergence to make conclusions about similar series.

Divergence of Series

A series diverges if the sum of its terms increases indefinitely as more terms are added. In mathematical analysis, determining the divergence of a series is crucial for understanding its behavior.
The key to identifying divergence is often through comparison with a known divergent series, like the harmonic series. Using comparison tests, such as the Limit Comparison Test, we can decide if a given series also diverges by seeing how its terms' limit compares to the terms of a divergent series.
For example, in the series \( \sum \frac{1}{n + \sqrt{n}} \), using the Limit Comparison Test with the harmonic series, we find that the ratio of the terms approaches 1 as \( n \) increases. This means the original series mimics the behavior of the harmonic series, confirming its divergence. Understanding divergence helps mathematicians know that no finite sum exists for such series, which influences how they are used in various math applications.