Convergent refers to a sequence or series that approaches a finite, specific value as the number of terms increases. In the context of calculus, convergence is a critical concept that determines the behavior and properties of infinite series and integrals.
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Convergent series have the property that the partial sums of the series approach a finite limit as the number of terms increases.
The Divergence Test and the Integral Test are two important methods used to determine whether a series is convergent or divergent.
Absolute convergence implies regular convergence, but the converse is not always true.
Convergent series have many desirable properties, such as the ability to rearrange the order of the terms without affecting the sum.
The Integral Test provides a way to compare the behavior of a series to the behavior of a related integral, which can be used to determine convergence or divergence.
Review Questions
Explain the relationship between convergent series and the Divergence Test.
The Divergence Test is a key tool used to determine whether a series is convergent or divergent. If the terms of a series do not approach zero as the index approaches infinity, then the series is divergent. Conversely, if the terms of a series do approach zero, then the series may be convergent, and other tests, such as the Integral Test, can be used to confirm convergence.
Describe how the Integral Test can be used to establish the convergence or divergence of a series.
The Integral Test allows you to compare the behavior of a series to the behavior of a related integral. If the integral converges, then the series also converges. If the integral diverges, then the series also diverges. This is a powerful tool because it can be easier to analyze the convergence or divergence of an integral than a series, especially for series where other tests may be inconclusive or difficult to apply.
Analyze the differences between regular convergence and absolute convergence, and explain why absolute convergence is a stronger form of convergence.
Regular convergence means that the series approaches a finite limit, but the absolute values of the terms may not converge. Absolute convergence, on the other hand, requires that the series formed by the absolute values of the terms also converges to a finite limit. Absolute convergence is a stronger form of convergence because it ensures that the series behaves well, even when the terms are rearranged. Absolute convergence implies regular convergence, but the converse is not always true, making absolute convergence a more desirable property for many applications.