A series is a special type of sequence where each term in the sequence is added together to form a sum. Series are often used to model and analyze various mathematical phenomena, particularly in the context of convergence and divergence.
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The Divergence Test and the Integral Test are two methods used to determine whether a series converges or diverges.
The Comparison Test compares a given series to a known series to determine the convergence or divergence of the given series.
The Divergence Test states that if the terms of a series do not approach 0 as the index approaches infinity, then the series diverges.
The Integral Test can be used to determine the convergence or divergence of a series by comparing it to the corresponding improper integral.
The Comparison Test states that if the terms of a series are less than or equal to the corresponding terms of a convergent series, then the given series converges.
Review Questions
Explain how the Divergence Test can be used to determine the convergence or divergence of a series.
The Divergence Test states that if the terms of a series do not approach 0 as the index approaches infinity, then the series diverges. This means that if the limit of the terms of the series as the index goes to infinity is not equal to 0, then the series will diverge. By examining the behavior of the terms of the series as the index increases, you can use the Divergence Test to determine whether the series will converge or diverge.
Describe how the Integral Test can be used to analyze the convergence or divergence of a series.
The Integral Test allows you to compare a series to the corresponding improper integral. If the improper integral converges, then the series also converges. If the improper integral diverges, then the series also diverges. This test is particularly useful for series where the terms are given by a continuous function, as the integral can be evaluated to determine the convergence or divergence of the series.
Explain how the Comparison Test can be used to determine the convergence or divergence of a series, and how it relates to the Divergence and Integral Tests.
The Comparison Test states that if the terms of a series are less than or equal to the corresponding terms of a convergent series, then the given series also converges. Conversely, if the terms of a series are greater than or equal to the corresponding terms of a divergent series, then the given series also diverges. This test allows you to leverage known information about the convergence or divergence of other series to make conclusions about the series in question. The Comparison Test is closely related to the Divergence and Integral Tests, as it provides an alternative way to analyze the behavior of a series.