Problem 22 - Thomas Calculus

Question

Which of the series in Exercises \(17-56\) converge, and which diverge? Use any method, and give reasons for your answers. $$ \sum_{n=1}^{\infty} \frac{n+1}{n^{2} \sqrt{n}} $$

Step-by-Step Solution

Answer

The series converges.

Step 1: Express the Series

Write the given series in a more manageable form for convergence tests:\[ \sum_{n=1}^{\infty} \frac{n+1}{n^{2} \sqrt{n}} = \sum_{n=1}^{\infty} \frac{n}{n^{2} \sqrt{n}} + \sum_{n=1}^{\infty} \frac{1}{n^{2} \sqrt{n}}. \]

Step 2: Simplify the Terms

Simplify each term of the series:\[ \frac{n}{n^{2} \sqrt{n}} = \frac{1}{n^{3/2}}, \quad \text{and} \quad \frac{1}{n^{2} \sqrt{n}} = \frac{1}{n^{5/2}}. \]

Step 3: Analyze Each Series Separately

Consider the series \( \sum_{n=1}^{\infty} \frac{1}{n^{3/2}} \) and \( \sum_{n=1}^{\infty} \frac{1}{n^{5/2}} \). Both are \( p \)-series with \( p = \frac{3}{2} \text{ and } \frac{5}{2}, \) respectively.

Step 4: Apply the p-Series Test

Recall that a \( p \)-series \( \sum_{n=1}^{\infty} \frac{1}{n^p} \) converges if \( p > 1 \). Since both \( \frac{3}{2} > 1 \) and \( \frac{5}{2} > 1 \), these series converge.

Step 5: Conclusion

Since both component series \( \sum_{n=1}^{\infty} \frac{1}{n^{3/2}} \) and \( \sum_{n=1}^{\infty} \frac{1}{n^{5/2}} \) converge, the original series \( \sum_{n=1}^{\infty} \frac{n+1}{n^{2} \sqrt{n}} \) also converges.

Key Concepts

Understanding p-Series

A "p-series" is a type of infinite series that takes the form \( \sum_{n=1}^{\infty} \frac{1}{n^p} \), where \( p \) is a constant exponent. p-Series are special because their convergence depends strictly on the value of \( p \).
When analyzing such a series, there's a simple rule: if \( p > 1 \), the series converges. If \( p \leq 1 \), the series diverges. This behavior is rooted in the "harmonic series," which is a classic example of a divergent p-series with \( p = 1 \).
Knowing this rule allows us to quickly determine the convergence of any p-series based solely on its exponent, making it a powerful tool in mathematical analysis.
Remember:

• If \( p > 1 \), expect convergence.
• If \( p \leq 1 \), prepare for divergence.

Exploring Convergence Tests

Convergence tests help us determine if a given series converges or diverges. One of the simplest and most reliable of these tests is the "p-Series Test." This test specifically applies to p-series, where you only need to look at the exponent \( p \) to decide the outcome.
There are other convergence tests too, like the "Comparison Test," "Integral Test," "Ratio Test," and "Root Test." Each test has its own conditions and strengths, suited to different types of series.
In the context of this exercise, the use of the p-Series Test was particularly effective because it matched the format required. By applying it, we confirmed that both component series of the given problem converge, making the entire series convergent.
**Key takeaway**: Choose the convergence test that best matches the structure of the series you're analyzing.

Series Simplification Techniques

The process of "series simplification" involves breaking down complex series into simpler parts to analyze them separately. Simplifying complicated expressions often reveals their true nature, making them easier to study.
In this exercise, the given series was split into two simpler series, each of which could be independently tested for convergence. By expressing the terms as \( \frac{1}{n^{3/2}} \) and \( \frac{1}{n^{5/2}} \), we easily recognized them as individual p-series.
This technique is useful because complex series might contain terms that behave differently when isolated, and simplification is the first step towards understanding these behaviors.
**Remember**:

• Simplify before you test.
• Break down complex problems into manageable parts.

Foundations of Mathematical Analysis

Mathematical analysis is a branch of mathematics that focuses on limits, functions, derivatives, series, and integrals. It provides the rigorous underpinning for many mathematical concepts, including the convergence of series.
Using principles from mathematical analysis, we can prove whether a series converges or not. It gives us the frameworks like the p-Series Test and other convergence tests, which act as tools for evaluating series.
In the given problem, mathematical analysis ensures that our conclusions about the series converge, allowing us to confidently apply theorems and rules like the p-Series Test.
Mathematical analysis isn't just about computation—it's about understanding why and how things work, providing a deep insight that pure calculation alone can't achieve.