Convergence tests for series from "summary" of Differential Calculus by S Balachandra Rao
Convergence tests are used to determine whether a given series converges or diverges. One such test is the comparison test, which states that if the terms of a series are all positive and can be shown to be smaller than the terms of a convergent series, then the original series must also converge. Another important test is the ratio test, which involves taking the limit of the ratio of consecutive terms in a series. If this limit is less than 1, then the series converges; if it is greater than 1, the series diverges; and if it is equal to 1, the test is inconclusive. The root test is similar to the ratio test, but involves taking the nth root of the absolute value of the terms in a series. If this limit is less than 1, then the series converges; if it is greater than 1, the series diverges; and if it is equal to 1, the test is inconclusive. The integral test is used for series whose terms are positive and decreasing. It involves comparing the series to an integral and determining whether the integral converges or diverges. If the integral converges, then the series also converges; if the integral diverges, then the series also diverges. Lastly, the alternating series test is used for series whose terms alternate in sign. If the terms of an alternating series decrease in absolute value and approach zero, then the series converges. This test is particularly useful for determining convergence of alternating series without having to compute the sum explicitly.- Convergence tests are essential tools in the realm of series, providing mathematicians with methods to determine the behavior of a series without having to compute its sum directly.
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