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 = 1 to ∞) k^(–1/5)

Verified step by step guidance

1. Identify the series given: \( \sum_{k=1}^{\infty} k^{-\frac{1}{5}} \). This is a series with terms of the form \( k^{-p} \), where \( p = \frac{1}{5} \).
2. Recognize that this is a p-series, which has the general form \( \sum_{k=1}^{\infty} \frac{1}{k^p} \). The convergence of a p-series depends on the value of \( p \).
3. Recall the p-series test: A p-series \( \sum \frac{1}{k^p} \) converges if and only if \( p > 1 \), and diverges otherwise.
4. Since \( p = \frac{1}{5} = 0.2 \), which is less than 1, the p-series test indicates that the series diverges.
5. Optionally, you could confirm this result using the Divergence Test by checking \( \lim_{k \to \infty} k^{-\frac{1}{5}} \). If the limit is not zero, the series diverges. Here, the limit is zero, so the Divergence Test is inconclusive, but the p-series test already gives the answer.

Key Concepts

Divergence Test

The Divergence Test states that if the limit of the terms of a series does not approach zero as k approaches infinity, 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 the convergence of a series to the convergence of an improper integral. If the function f(k) = a_k is positive, continuous, and decreasing for k ≥ 1, then the series ∑a_k converges if and only if the integral from 1 to infinity of f(x) dx converges.

p-series Test

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