Final answer:
The integral test helps determine the convergence of a series by comparing it to the convergence of an improper integral. A series converges if and only if the corresponding integral does, provided the function is continuous, positive, and decreasing.
Step-by-step explanation:
The question refers to the integral test for convergence of series in mathematics. The integral test states that if a function f is continuous, positive, and decreasing on the interval [1, ∞), then the series ∑f(n) converges if and only if the improper integral ∫f(x)dx from 1 to ∞ converges. When you take the limit as the step size approaches zero in a Riemann sum of f(x), and the number of steps approaches infinity, you are effectively using the integral test by replacing the summation with an integral to determine the convergence of the series.
To clarify, the function ceasing to be continuous, positive, and decreasing would invalidate the use of the integral test. This is because the integral test relies on the assumption that the function is monotonically decreasing to ensure that the sum of the function's values over the integers is comparable to the integral of the function over the corresponding continuous interval.