Problem 10 - Thomas Calculus

Question

Use the Limit Comparison Test to determine if each series converges or diverges. \begin{equation}\begin{array}{l}{\sum_{n=1}^{\infty} \sqrt{\frac{n+1}{n^{2}+2}}} \\ {\quad\left(\text {Hint} : \text { Limit Comparison with } \sum_{n=1}^{\infty}(1 / \sqrt{n})\right)}\end{array}\end{equation}

Step-by-Step Solution

Answer

The series diverges.

Step 1: Identify Terms for Limit Comparison

Given the series \( \sum_{n=1}^{\infty}\sqrt{\frac{n+1}{n^{2}+2}} \), we need to compare it with a simpler series. The hint suggests using the series \( \sum_{n=1}^{\infty}\frac{1}{\sqrt{n}} \) for the limit comparison test.

Step 2: Setup the Limit for Comparison

We will find the limit of the ratio of the terms of the given series and the series from the hint. Let \( a_n = \sqrt{\frac{n+1}{n^{2}+2}} \) and \( b_n = \frac{1}{\sqrt{n}} \). Compute the limit: \[L = \lim_{n \to \infty} \frac{a_n}{b_n} = \lim_{n \to \infty} \frac{\sqrt{\frac{n+1}{n^{2}+2}}}{\frac{1}{\sqrt{n}}} = \lim_{n \to \infty} \sqrt{\frac{n(n+1)}{n^2 + 2}}.\]

Step 3: Simplify the Limit Expression

Simplify the expression within the limit:\[L = \lim_{n \to \infty} \sqrt{\frac{n^2 + n}{n^2 + 2}}.\]Divide the numerator and the denominator by \( n^2 \):\[L = \lim_{n \to \infty} \sqrt{\frac{1 + \frac{1}{n}}{1 + \frac{2}{n^2}}}.\]

Step 4: Evaluate the Limit

As \( n \to \infty \), \( \frac{1}{n} \to 0 \) and \( \frac{2}{n^2} \to 0 \). Therefore, the expression simplifies to:\[L = \sqrt{\frac{1 + 0}{1 + 0}} = \sqrt{1} = 1.\]

Step 5: Apply the Limit Comparison Test Conclusion

The limit \( L = 1 \) is a positive finite number. Thus, by the Limit Comparison Test, both series \( \sum_{n=1}^{\infty}\sqrt{\frac{n+1}{n^{2}+2}} \) and \( \sum_{n=1}^{\infty}\frac{1}{\sqrt{n}} \) share the same behavior regarding convergence or divergence.

Step 6: Determine the Behavior of the Comparison Series

The series \( \sum_{n=1}^{\infty}\frac{1}{\sqrt{n}} \) diverges because it is a \( p \)-series with \( p = \frac{1}{2} \), where \( p \leq 1 \). Therefore, the original series diverges as well.

Key Concepts

Understanding Convergence and Divergence

One of the foundational concepts in calculus series is understanding whether a series converges or diverges. A series converges when the sum of its terms approaches a specific, finite value as more terms are added. This means that as you keep adding more terms, you are getting closer to a certain number.
In contrast, a series diverges if its sum grows indefinitely, or does not approach any finite limit. Determining these properties helps us understand the behavior of mathematical expressions and their sums as they extend towards infinity.
The Limit Comparison Test is particularly useful for evaluating the convergence or divergence of a series by comparing it with another series whose behavior we can easily assess. If the limit of the ratio of two series' terms is a positive finite number, both series will either converge or diverge together. This means that if one diverges, so will the other, and the same goes for convergence.
Overall, knowing whether a series converges or diverges is crucial for applications in many areas of mathematics and science, guiding us in the evaluation of infinite sums.

Exploring the P-Series

In the study of calculus series, a p-series is a specific type of series with the form:

• \( \sum_{n=1}^{\infty} \frac{1}{n^p} \)

Where \( p \) is a positive constant. The convergence of a p-series depends heavily on the value of \( p \). A key rule to remember is:

• When \( p > 1 \), the series converges.
• When \( p \leq 1 \), the series diverges.

In the exercise above, the series \( \sum_{n=1}^{\infty} \frac{1}{\sqrt{n}} \) is a p-series with \( p = \frac{1}{2} \). Since \( \frac{1}{2} \leq 1 \), this series diverges.
The simplicity of p-series makes them a frequent choice for using the Limit Comparison Test. They're often easier to analyze than complex or compounded fractions, making them a key tool in understanding series behavior.

Digging Deeper into Calculus Series

Calculus series are central to advanced mathematics. Understanding them helps in solving complex problems across physics, engineering, economics, and beyond. These series take many forms, but at their core, they build on the principle of summing sequences of numbers.
Series are widely used because they help approximate functions and solve differential equations. The broader concept of a series extends beyond just adding numbers; it explores the behavior of the sum as the series progresses. Calculus series can be geometric, arithmetic, harmonic, or more complex types like p-series.
For series analysis, tests like the Limit Comparison Test become vital. They enable us to methodically determine whether a series will keep growing indefinitely or settle towards a finite sum. The resulting insights are foundational for continuous functions and real-world applications, underlining the importance of grasping these fundamental concepts in calculus.