What is the connection between improper integrals, Riemann sums, and the integral test?
The connection between improper integrals, Riemann sums, and the integral test lies in their relation to evaluating the convergence or divergence of certain types of infinite series or improper integrals.
1. Improper integrals: An improper integral arises when one or both of the bounds of integration are infinite or when the function being integrated has a singularity within the interval of integration. For instance, the integral from 1 to infinity of 1/x^n, where n is a positive real number, is an improper integral. To determine if an improper integral converges or diverges, we often use the integral test.
2. Riemann sums: Riemann sums are a method used to approximate the value of a definite integral by dividing the interval of integration into smaller subintervals and calculating the areas of corresponding rectangles (or other shapes) under the curve. The idea behind Riemann sums is that as the number of subintervals increases, the approximation becomes more accurate and approaches the exact value of the integral.
3. Integral test: The integral test, or the integral criterion, is a convergence test that compares the convergence of an infinite series to the convergence of an associated improper integral. Specifically, if a series has terms that are positive, decreasing, and can be expressed in the form of f(n), where f(x) is a continuous, positive, and decreasing function on the interval [1,∞], then the series converges if and only if the corresponding improper integral converges.
To apply the integral test, you need to follow these steps:
1. Verify that the series has terms that are positive and decreasing.
2. Identify a function f(n) that describes the terms of the series.
3. Determine if the function f(x) is continuous, positive, and decreasing on the interval [1,∞].
4. Evaluate the corresponding improper integral by integrating the function over the same interval.
5. If the integral converges, then the series converges. If the integral diverges, then the series diverges.
In summary, improper integrals help us evaluate the convergence or divergence of certain functions, while Riemann sums are a technique used to approximate definite integrals. The integral test allows us to determine the convergence of series by comparing them to associated improper integrals.