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In this section we will discuss using the alternating series test to determine if an infinite series converges or diverges It turns out that there is a powerful test for determining that a series of this form converges. The alternating series test can be used only if the terms of the series alternate in sign
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A proof of the alternating series test is also given. In particular, we are interested in series whose terms alternate between positive and negative (aptly named alternating series) In this section we introduce alternating series—those series whose terms alternate in sign
We will show in a later chapter that these series often arise when studying power series.
The alternating series test (or also know as the leibniz test) helps us determine whether a given alternating series is convergent or not In this article, we’ll learn what type of series will benefit from the alternating series test. In mathematical analysis, the alternating series test proves that an alternating series is convergent when its terms decrease monotonically in absolute value and approach zero in the limit. Clearly, to show the convergence, you need to check all these three conditions
What can you conclude if a given alternating series fails one or more of the three conditions Unfortunately, the failure of this test does not immediately lead to the divergence! Discover the key conditions and rules for applying the alternating series test Learn when to use this powerful tool and how it compares to other convergence tests in calculus.
The following theorem provides a bound for the size of this error for series that satisfy the conditions of the alternating series test
Specifically, the error is smaller than bn+1, the absolute value of the first neglected term. Series (2), shown as the second alternating series example, is called the alternating harmonic series We will show that whereas the harmonic series diverges, the alternating harmonic series converges.
