Integral Test
Definition
The integral test is a method used to determine the convergence or divergence of an infinite series by comparing it to the convergence or divergence of an improper integral.
Analogy
Think of the integral test as a way to measure how much water is in a swimming pool. By comparing the amount of water in the pool (the series) to the capacity of the pool (the integral), we can determine if it will overflow (divergent) or stay within limits (convergent).
Related terms
Convergence: The property of a series that indicates it approaches a finite value as more terms are added.
Divergence: The property of a series that indicates it does not approach a finite value as more terms are added.
Improper Integral: An integral with one or both limits being infinite, or having an integrand that becomes unbounded at some point within its interval.
"Integral Test" appears in:
Study guides (2)
Additional resources (1)
Practice Questions (11)
Which statement accurately describes the Integral Test for convergence?
Which condition is NOT necessary for the Integral Test to be applicable?
Given a series ∑(n=1 to ∞) (1/n^3), what can be concluded about its convergence using the Integral Test?
Which of the following series can be evaluated using the Integral Test?
Given that ∫(n=1 to ∞) (1/(n^2 + 1)) converges, for which of the following series does the Integral Test guarantee convergence?
Which of the following series diverges according to the Integral Test?
For a series to be analyzed using the Integral Test, which property must the terms of the series satisfy?
Which of the following statements accurately describes the convergence behavior of a series when the corresponding integral is inconclusive according to the Integral Test?
Which of the following series converges according to the Integral Test?
What is the convergence behaviour of the series ∑(n=1 to ∞) (1/1+n) based on the Integral Test?
What is the convergence behavior of the series ∑(n=1 to ∞) 4/(n^2 +1) based on the Integral Test?
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