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: \( f(x) = \frac{1}{\sqrt[3]{5x + 3}} \). This function is defined for \( x \geq 1 \).
2. Check the conditions for the Integral Test: verify that \( f(x) \) is positive, continuous, and decreasing for \( x \geq 1 \). Since the cube root function is continuous and increasing, and the denominator \( 5x + 3 \) is increasing, \( f(x) \) is positive and continuous. To check if \( f(x) \) is decreasing, consider the derivative \( f'(x) \) and verify it is negative for \( x \geq 1 \).
3. Set up the improper integral corresponding to the series: \( \int_1^{\infty} \frac{1}{\sqrt[3]{5x + 3}} \, dx \).
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 improper 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, the series diverges as well.

Key Concepts

Integral Test for Convergence

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, then the series converges if and only if the integral of the function from a certain point to infinity converges.

Conditions for Applying the Integral Test

To apply the Integral Test, the function f(x) representing the series terms must be positive, continuous, and decreasing for all x greater than or equal to some number 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 involves integrating the function from a finite point to infinity and determining if the limit exists and is finite. This process often requires substitution and limit evaluation techniques to conclude whether the integral converges or diverges.