48–63. Choose your test Determine whether the following series converge or diverge using the properties and tests introduced in Sections 10.3 and 10.4.
2 / 4² + 2 / 5² + 2 / 6² + ⋯

Verified step by step guidance

1. Identify the series given: it is \( \frac{2}{4^2} + \frac{2}{5^2} + \frac{2}{6^2} + \cdots \). This is an infinite series where the general term can be written as \( a_n = \frac{2}{n^2} \) starting from \( n=4 \).
2. Recognize the type of series: since the terms involve \( \frac{1}{n^2} \), this resembles a p-series, which has the general form \( \sum \frac{1}{n^p} \). Here, \( p = 2 \).
3. Recall the p-series test: a p-series \( \sum \frac{1}{n^p} \) converges if and only if \( p > 1 \). Since \( p = 2 > 1 \), the series \( \sum \frac{1}{n^2} \) converges.
4. Since the series has a constant multiple of 2, factor it out: \( \sum_{n=4}^\infty \frac{2}{n^2} = 2 \sum_{n=4}^\infty \frac{1}{n^2} \). Multiplying a convergent series by a constant does not affect convergence.
5. Conclude that the given series converges by the p-series test because it is a constant multiple of a convergent p-series with \( p=2 \).

Key Concepts

Convergence and Divergence of Series

A series converges if the sum of its terms approaches a finite limit as the number of terms increases indefinitely. If the sum grows without bound or oscillates, the series diverges. Understanding this distinction is fundamental to analyzing infinite series.

p-Series Test

The p-series test determines convergence based on the form ∑ 1/n^p. If p > 1, the series converges; if p ≤ 1, it diverges. This test is useful for series with terms involving powers of n, like the given series with terms 2/n².

Comparison Test

The comparison test involves comparing a given series to a known benchmark series to determine convergence or divergence. If the terms of the given series are smaller than those of a convergent series, it also converges; if larger than a divergent series, it diverges.