Calc1231W11T2 - The ratio test

Sunday, 14 October 2018

4:10 PM

Untitled picture.png The ratio test is a simple, but sometimes inconclusive; critera to check the convergence or divergence of an infinite series. The Ratio Test: Suppose that E an is a series of real numbers, and that the limit an +1 r = lim exists Then • ifr < 1 then an is convergent, • ifr > 1 then an is divergent, • if r = 1 the test is inconclusive For example EDO an where an = 5 is a series where the limit lim is less than one. So by the ratio test the series is convergent In this case the ratio test guarantees the existence of the limit I; But it gives no information as to the value of L. For this, we can use properties of a geometric series to evaluate L = 1/5/(1-1/5) Untitled picture.png Machine generated alternative text: The ratio test tells you about the convergence or divergence of a series, provided the limiting ratio lim an r exists and is not equal to 1. If r series may converge or diverge For example consider the series E an where an n=l Because an + 1 n/(n+l) then an +1 r = lim 1 —1. In this case, the ratio test has unfortunately given no information as to the convergence or divergence of this series Now consider the series bn where bn n=l Because 1 bn then Again, the ratio test tells us nothing about the convergence or divergence of this series. bn So we see that the ratio test really is inconclusive. Both examples had the same r value but the first example diverges, by the p-test 1 the test is inconclusive and the O and the second example converges, by the p-test O . In fact 2 6 Ink Drawings Ink Drawings  
Untitled picture.png Machine generated alternative text: The ratio test tells you about the convergence or divergence of a series, provided the limiting ratio lim an r exists and is not equal to 1. If r series may converge or diverge For example consider the series E an where an n=l Because an + 1 n/(n+l) then an +1 r = lim 1 —1. In this case, the ratio test has unfortunately given no information as to the convergence or divergence of this series Now consider the series bn where bn n=l Because 1 bn then Again, the ratio test tells us nothing about the convergence or divergence of this series. bn So we see that the ratio test really is inconclusive. Both examples had the same r value but the first example diverges, by the p-test 1 the test is inconclusive and the O and the second example converges, by the p-test O . In fact 2 6 Untitled picture.png One of the most important parts of using the ratio test for the series E Find the value of r for the the following series: an is calculating an +1 r = lim • ifan • ifan 5 • ifan then r • ifan then r n! then r then r Untitled picture.png The following examples have been taken from sample final exams, which are available from the UNSW library MATH1231 2012s2 Q4iib) Determine whether each of the following series converges or diverges; stating any tests you use Answer: let an = n4/n! Since lim n=l an + 1 an the series converges O by the ratio test MATH1231 2013s2 Q4ia) Use appropriate tests to determine whether each of the following series converges or diverges: n=l Answer: let an Since O by the ratio test lim an + 1 an the series converges MATH1231 2014s2 Qliiib) Determine whether each of the following series converges or diverges, stating any tests you use: Answer: let an 2 Since 2n + 3n lim an + 1 an the series converges O by the ratio test
Untitled picture.png The following examples have been taken from sample final exams, which are available from the UNSW library MATH1231 2012s2 Q4iib) Determine whether each of the following series converges or diverges; stating any tests you use Answer: let an = n4/n! Since lim n=l an + 1 an the series converges O by the ratio test MATH1231 2013s2 Q4ia) Use appropriate tests to determine whether each of the following series converges or diverges: n=l Answer: let an Since O by the ratio test lim an + 1 an the series converges MATH1231 2014s2 Qliiib) Determine whether each of the following series converges or diverges, stating any tests you use: Answer: let an 2 Since 2n + 3n lim an + 1 an the series converges O by the ratio test Untitled picture.png One of the most common uses of the ratio test is to find values of x such that the series converges, where an is a function of an For example, if this series will converge by the ratio test, provided lim Let's rephrase this as an inequality in Since an +1 then this series will converge by the ratio test provided satisfies the inequality: abs(x) < 5 Note: the Maple syntax for the inequality < 10 is abs (x) < 10. an +1 an
Untitled picture.png One of the most common uses of the ratio test is to find values of x such that the series converges, where an is a function of an For example, if this series will converge by the ratio test, provided lim Let's rephrase this as an inequality in Since an +1 then this series will converge by the ratio test provided satisfies the inequality: abs(x) < 5 Note: the Maple syntax for the inequality < 10 is abs (x) < 10. an +1 an

 

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