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The Integral Test and its Applications in Calculus

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The Integral Test is a calculus theorem used to determine the convergence of infinite series by comparing them to improper integrals. It requires the function to be continuous, positive, and decreasing on [1, ∞). This test is crucial for series that are difficult to sum, such as ∑(1/n^2), and is complemented by the Comparison Test for assessing convergence.

Understanding the Integral Test for Series Convergence

The Integral Test is an essential theorem in calculus that establishes a criterion for determining the convergence or divergence of an infinite series. Developed by Augustin-Louis Cauchy, the Integral Test compares an infinite series to an improper integral of a corresponding function. For the test to be applicable, the function must be continuous, positive, and decreasing on the interval [1, ∞). If the improper integral of the function is finite, the series converges; conversely, if the integral is infinite, the series diverges. This test is particularly useful for series whose terms are given by a function that is difficult to sum explicitly.
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Applying the Integral Test Step by Step

To apply the Integral Test, one must first ensure that the series under consideration can be represented by a function that is continuous, positive, and decreasing on the interval [1, ∞). After identifying such a function, the next step is to set up an improper integral with the same limits. Evaluating this integral will determine the convergence or divergence of the series. For example, the series ∑(1/n^2) corresponds to the function f(x) = 1/x^2, which meets the criteria. The improper integral from 1 to infinity of this function converges, hence the series converges as well.

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Criteria for applying the Integral Test

Function must be continuous, positive, decreasing on [1, ∞).

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Convergence indication via Integral Test

If improper integral of function is finite, series converges.

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Divergence indication via Integral Test

If improper integral of function is infinite, series diverges.

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