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))

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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.

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 \( \frac{\sqrt{n} + 1}{n} \).

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.

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 \( p > 1 \).

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.

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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.

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Direct Comparison Test

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.

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Intro to Series: Partial Sums

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.

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