Why The Ratio Test Stays Hidden Behind Every Great Math Mystery - hub.bocatc.org
Most people learn the Ratio Test as a formula without understanding the reasoning behind it. Even if you took real analysis, the standard proof might have left you more confused than enlightened. In this section we will discuss using the Ratio Test to determine if an infinite series converges absolutely or diverges. The Ratio Test can be used on any series, but unfortunately will not always yield a conclusive answer as to whether a series will converge absolutely or diverge. A proof of the Ratio Test is also given. @GCab That's correct, the ratio test doesn't apply to the harmonic series. But the statement I quoted is still invalid.
The ratio test is best used when you have certain elements in the sum. The way to get a feel for this is to build a set of tables containing examples of tests that work as you are working practice problems. This is an extremely powerful technique that will help you really understand infinite series. Here is a list of things to watch for. The ratio test is a most useful test for series convergence. It caries over intuition from geometric series to more general series. Learn more about it here. If the ratio test gives a limit of , then the series is like a geometric series of ratio . The case of is an edge case, and can go either way, which we will demonstrate with specific examples in a bit. Now that you have the basic idea, we give examples showing the following. Convergence by the ratio test. Divergence by the ratio test. Decreasing is not enough. In the language of mathematics, it's necessary but not sufficient: the sum can't converge if the terms are getting bigger and bigger. But getting smaller doesn't guarantee convergence. The harmonic series 1 + 1/2 + 1/3 + 1/4 +... is a famous counterexample of a series with terms that are decreasing but that doesn't ... The ratio test for convergence lets us determine the convergence or divergence of a series a_n using a limit, L. Once we find a value for L, the ratio test tells us that the series converges absolutely if L<1, and diverges if L>1 or if L is infinite. The test is inconclusive if L=1. The ratio test is used most often when our series includes a factorial or something raised to the nth power. In this section, we prove the last two series convergence tests: the ratio test and the root test. These tests are nice because they do not require us to find a comparable series. The ratio test will.
The ratio test for convergence lets us determine the convergence or divergence of a series a_n using a limit, L. Once we find a value for L, the ratio test tells us that the series converges absolutely if L<1, and diverges if L>1 or if L is infinite. The test is inconclusive if L=1. The ratio test is used most often when our series includes a factorial or something raised to the nth power. In this section, we prove the last two series convergence tests: the ratio test and the root test. These tests are nice because they do not require us to find a comparable series. The ratio test will.
