Direct Comparison Test
In Exercises 1–8, use the Direct Comparison Test to determine if each series converges or diverges.
∑ (from n=1 to ∞) cos²n / n^(3/2)

Verified step by step guidance

1. Identify the given series: \( \sum_{n=1}^{\infty} \frac{\cos^{2} n}{n^{3/2}} \). Notice that the terms involve \( \cos^{2} n \) divided by \( n^{3/2} \).
2. Recall the Direct Comparison Test: To use it, we need to compare the given series to a known benchmark series \( \sum b_n \) where \( b_n \) is positive and whose convergence behavior is known.
3. Since \( \cos^{2} n \) oscillates between 0 and 1, we know \( 0 \leq \cos^{2} n \leq 1 \) for all \( n \). Therefore, each term of the series satisfies \( 0 \leq \frac{\cos^{2} n}{n^{3/2}} \leq \frac{1}{n^{3/2}} \).
4. Compare the given series to the p-series \( \sum_{n=1}^{\infty} \frac{1}{n^{3/2}} \). Since \( p = \frac{3}{2} > 1 \), this p-series converges.
5. By the Direct Comparison Test, since \( 0 \leq \frac{\cos^{2} n}{n^{3/2}} \leq \frac{1}{n^{3/2}} \) and \( \sum \frac{1}{n^{3/2}} \) converges, the original series \( \sum \frac{\cos^{2} n}{n^{3/2}} \) also converges.

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 0 ≤ a_n ≤ b_n for all n and ∑b_n converges, then ∑a_n also converges. Conversely, if a_n ≥ b_n ≥ 0 and ∑b_n diverges, then ∑a_n diverges.

Behavior of the Cosine Function

The term cos²(n) oscillates between 0 and 1 for all integer n. Since cos²(n) is always non-negative and bounded above by 1, it can be used to compare the given series to a simpler series by replacing cos²(n) with its upper bound.

p-Series Test

A p-series ∑ 1/n^p converges if and only if p > 1 and diverges otherwise. In this problem, the denominator n^(3/2) corresponds to p = 3/2, which is greater than 1, so the p-series converges. This fact helps in comparing the given series to a convergent p-series.