17–22. Integral Test Use the Integral Test to determine whether the following series converge after showing that the conditions of the Integral Test are satisfied.
∑ (k = 1 to ∞) 1 / (∛(5k + 3))

Verified step by step guidance

1. First, identify the function corresponding to the terms of the series: define \( f(x) = \frac{1}{\sqrt[3]{5x + 3}} \). This function will be used for the Integral Test.
2. Check the conditions for the Integral Test: verify that \( f(x) \) is positive, continuous, and decreasing for \( x \geq 1 \). Since the denominator \( \sqrt[3]{5x + 3} \) is positive and increasing, \( f(x) \) is positive and decreasing on this interval.
3. Set up the improper integral to test convergence: consider \( \int_1^{\infty} \frac{1}{\sqrt[3]{5x + 3}} \, dx \).
4. Evaluate the integral by using an appropriate substitution, such as \( u = 5x + 3 \), which simplifies the integral to a form involving \( u^{-1/3} \).
5. Determine whether the integral converges or diverges by evaluating the limit as the upper bound approaches infinity. If the integral converges, then by the Integral Test, the series \( \sum_{k=1}^\infty \frac{1}{\sqrt[3]{5k + 3}} \) also converges; if it diverges, so does the series.

Key Concepts

Integral Test

The Integral Test determines the convergence of an infinite series by comparing it to an improper integral. If the function corresponding to the series terms is positive, continuous, and decreasing for all x ≥ N, then the series and the integral either both converge or both diverge.

Conditions for Applying the Integral Test

To use the Integral Test, the function f(x) representing the series terms must be positive, continuous, and decreasing on the interval [N, ∞). Verifying these conditions ensures the test's validity and that the behavior of the integral reflects the series' behavior.

Evaluating Improper Integrals

Evaluating the improper integral ∫ from N to ∞ of f(x) dx involves taking limits as the upper bound approaches infinity. The convergence or divergence of this integral directly informs the convergence of the corresponding series under the Integral Test.