Direct Comparison Test
In Exercises 1–8, use the Direct Comparison Test to determine if each series converges or diverges.
∑ (from n=1 to ∞) (√n + 1) / (√(n² + 3))

Verified step by step guidance

1. Identify the given series: \( \sum_{n=1}^{\infty} \frac{\sqrt{n} + 1}{\sqrt{n^2 + 3}} \). We want to determine if it converges or diverges using the Direct Comparison Test.
2. Simplify the general term to understand its behavior for large \( n \). Notice that \( \sqrt{n^2 + 3} \) behaves like \( n \) as \( n \to \infty \), so the term roughly behaves like \( \frac{\sqrt{n} + 1}{n} \).
3. Split the term into two parts: \( \frac{\sqrt{n}}{n} + \frac{1}{n} = \frac{1}{\sqrt{n}} + \frac{1}{n} \). This helps us compare with known series.
4. Choose appropriate comparison series: compare \( \frac{1}{\sqrt{n}} \) with \( \sum \frac{1}{n^{1/2}} \) and \( \frac{1}{n} \) with the harmonic series \( \sum \frac{1}{n} \). Recall that \( \sum \frac{1}{n^{p}} \) converges if and only if \( p > 1 \).
5. Use the Direct Comparison Test by comparing the original term to these simpler terms. Since \( \frac{1}{\sqrt{n}} \) diverges and dominates the behavior, conclude about the convergence or divergence of the original series accordingly.

Key Concepts

Direct Comparison Test

The Direct Comparison Test determines the convergence or divergence of a series by comparing it to another series with known behavior. If the terms of the given series are less than or equal to the terms of a convergent series, it also converges. Conversely, if the terms are greater than or equal to those of a divergent series, it diverges.

Behavior of Series Terms for Large n

Analyzing the behavior of the series terms as n approaches infinity helps simplify complex expressions. For example, dominant terms in numerator and denominator guide the comparison to simpler series like p-series or geometric series, facilitating the application of convergence tests.

p-Series and Their Convergence

A p-series has the form ∑ 1/n^p and converges if p > 1, diverging otherwise. Recognizing or approximating a given series to a p-series allows for straightforward conclusions about convergence, which is essential when using the Direct Comparison Test.