Calc1231W11T3 - Convergence of series with positive and negative terms

Sunday, 14 October 2018

4:40 PM

A series al + a2 + • is called alternating if the product of ftvo consecutive terms is negative, that is an x an +1 < 0. Or; in other words, consecutive terms have different sigm Which of the following series are alternating? 2 1 1 1 1

 

When does a series converge? This can be a tricky problenm Even for an alternating series; this is not so obvious. The following series baffled many 18th century mathematicians: Does it converge? One school of thought argued that we could arrange the terms of the series as follows so that the infinite sum is so that the infinite sum is O O Others argued we could equally well arrange it in the form Others argued for a compromise using the partial sum sn of the first n terms: 1 n is odd, 0 n is even. The infinite sum should be the average value of sn, othepwise known as the Cesäro sum; which is Sl +82+ In this course, according to our definition of convergence, this series diverges Ernesto Cesäro: 1859 - 1906_

 

In the 18th century Gottried Leibniz (the same of Calculus fame) came up with a simple criterion called the Alternating series test that applies to particular kinds of alternating series Theorem: Alternating series test Suppose that al , a,2, (13, • is a sequence of real numbers such that i) an > 0 for all n e Z + which is to say the numbers an are all positive, ii) an 2 an +1 for all n e Z + which is to say the numbers an are monotonically decreasing, which means decreasing or equal, iii) lim an = 0 which is to say the numbers an approach zero. Then the alternating series converges. For example, consider the alternating series E = al — a2 -F (13 — — + 1 1 1 32 33 Let an 1/3"n o - Then i) this is a sequence of positive numbers: an > o ii) this is a monotonically decreasing sequence, but proving this usually requires a little work The best place to start working is from a statement we can all agree upon, such as 1 Multiplying both sides of this inequality by 1,'3An iii) finally 0 gives 1 lim an 1

 

Hence the series converges by Leibniz's Alternating series test. (-1)'1+1an 1 3 1 32 1 33 Gottfried Wilhelm von Leibniz: 1646 - 1716

 

Suppose the series of positive and negative terms (1) converges. There are ftvo types of convergence for this series, which are distinguished by the behaviour of series of positive terms lall+ + + la41+• (2) If series (1) converges but series (2) diverges then we say that series (1) is conditionally convergent. Otherwise, when series (1) and (2) are both convergent we say that series (1) is absolutely convergent. Types of convergence • IS • • diverges, we say that 1 Convergent Convergent For example, consider the series Divergent Convergent Conditionally Convergent Absolutely Convergent (—1!1+1 n=l which is an alternating version of the famous harmonic series. 1 1 3 1 4 1 1 To prove the series 1 — — + Let an • • converges, we use Leibniz's Alternating series test o - Then i) the terms an, n e Z + are all positive: an > 0, ii) the terms are monotonically decreasing This is because the sequence 1, 2, 3, • monotonically decreasing ' lim an O monotonically increasing ' and the reciprocal of a monotonically increasing sequence is Hence by Leibniz's Alternating series test, the series converges. Because the harmonic series 1 + convergent; it is conditionally convergent 1 1 4 + • is more than just

 

In the previous question we saw two types of convergence. In contrast; there is only one type of divergence. Theorem: Suppose an is a sequence of real numbers (positve or negative). If the series n=l is divergent, then the series is divergent. Let's append this to our table, which gives a complete picture of the types of convergence and divergence. Types of convergence and divergence Series an is Series lanl is Series an is called Divergent Convergent Convergent For example, the series Divergent Divergent Convergent 1 2 1 22 Divergent Conditionally Convergent Absolutely Convergent 1 1 1 23 24 25 is not alternating, so Leibniz's alternating series test can not be applied However the series of positive terms is a geometric series, and converges to 2 22 23 24 25 an is Then using the table above, the series absolutely convergent

 

 

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