23–38. Divergence, Integral, and p-series Tests Use the Divergence Test, the Integral Test, or the p-series test to determine whether the following series converge.
∑ (k = 3 to ∞) 1 / (k − 2)⁴

Verified step by step guidance

1. Identify the given series: \( \sum_{k=3}^{\infty} \frac{1}{(k-2)^4} \). Notice that the term inside the summation can be rewritten as \( \frac{1}{n^4} \) by letting \( n = k - 2 \). This shifts the index to start from \( n=1 \).
2. Recognize that the series is a p-series of the form \( \sum_{n=1}^{\infty} \frac{1}{n^p} \) where \( p = 4 \). The p-series test states that such a series converges if and only if \( p > 1 \).
3. Since \( p = 4 > 1 \), the p-series test indicates that the series converges.
4. Optionally, you could apply the Integral Test by considering the function \( f(x) = \frac{1}{x^4} \) for \( x \geq 1 \), which is positive, continuous, and decreasing. Then evaluate the improper integral \( \int_1^{\infty} \frac{1}{x^4} \, dx \) to confirm convergence.
5. Conclude that by the p-series test (and optionally the Integral Test), the series \( \sum_{k=3}^{\infty} \frac{1}{(k-2)^4} \) converges.

Key Concepts

Divergence Test

The Divergence Test states that if the limit of the terms of a series does not approach zero, the series diverges. It is a quick initial check to determine if a series cannot converge, but if the limit is zero, the test is inconclusive.

Integral Test

The Integral Test relates a series to an improper integral by comparing the sum of terms to the integral of a corresponding function. If the integral converges, so does the series; if the integral diverges, the series diverges as well. It requires the function to be positive, continuous, and decreasing.

p-series Test

A p-series is a series of the form ∑ 1/n^p. It converges if and only if p > 1 and diverges otherwise. This test is useful for quickly determining convergence of series with terms involving powers of n.