#Tricky test 2 infinity calculator series#
įor ∑ n = 1 ∞ a n ∑ n = 1 ∞ a n with nonnegative terms, compare with a known series ∑ n = 1 ∞ b n. If | r | 1, p > 1, the series converges.įor p = 1, p = 1, we have the harmonic series ∑ n = 1 ∞ 1 / n. If lim n → ∞ a n ≠ 0, lim n → ∞ a n ≠ 0, the series diverges. This test cannot prove convergence of a series. If lim n → ∞ a n = 0, lim n → ∞ a n = 0, the test is inconclusive. If this test does not provide any information, try the integral test.įor any series ∑ n = 1 ∞ a n, ∑ n = 1 ∞ a n, evaluate lim n → ∞ a n. Do the terms in the series contain a factorial or power? If the terms are powers such that a n = b n n, a n = b n n, try the root test first.Is the series similar to a p − series p − series or geometric series? If so, try the comparison test or limit comparison test.If we are interested in absolute convergence, proceed to step 3, 3, considering the series of absolute values ∑ n = 1 ∞ | a n |. Is it an alternating series? Are we interested in absolute convergence or just convergence? If we are just interested in whether the series converges, apply the alternating series test.Is ∑ n = 1 ∞ a n ∑ n = 1 ∞ a n a familiar series? For example, is it the harmonic series (which diverges) or the alternating harmonic series (which converges)? Is it a p − series p − series or geometric series? If so, check the power p p or the ratio r r to determine if the series converges.In the steps below, we outline a strategy for determining whether the series converges. Problem-Solving Strategy: Choosing a Convergence Test for a SeriesĬonsider a series ∑ n = 1 ∞ a n. Here we introduce the ratio test, which provides a way of measuring how fast the terms of a series approach zero. The series ∑ n = 1 ∞ 1 / n ∑ n = 1 ∞ 1 / n diverges because the terms in the sequence do not approach zero fast enough as n → ∞. However, only the series ∑ n = 1 ∞ 1 / n 2 ∑ n = 1 ∞ 1 / n 2 converges. We know that 1 / n → 0 1 / n → 0 and 1 / n 2 → 0. For example, consider the series ∑ n = 1 ∞ 1 / n ∑ n = 1 ∞ 1 / n and the series ∑ n = 1 ∞ 1 / n 2. Not only do we need a n → 0, a n → 0, but we need a n → 0 a n → 0 quickly enough. From our earlier discussion and examples, we know that lim n → ∞ a n = 0 lim n → ∞ a n = 0 is not a sufficient condition for the series to converge. Ratio TestĬonsider a series ∑ n = 1 ∞ a n. Therefore, at the end of this section we discuss a strategy for choosing which convergence test to use for a given series. Throughout this chapter, we have seen that no single convergence test works for all series. The ratio test will be especially useful in the discussion of power series in the next chapter. These tests are particularly nice because they do not require us to find a comparable series. In this section, we prove the last two series convergence tests: the ratio test and the root test. 5.6.3 Describe a strategy for testing the convergence of a given series.5.6.2 Use the root test to determine absolute convergence of a series.5.6.1 Use the ratio test to determine absolute convergence of a series.
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