Lecture 17 The Nth term test and the comparison test

Lecture 17: The N th term test and the comparison test 2011 1 Introduction We are in the process of studying methods for concluding when a series (that is, an infinite summation) converges (that is, is equal to a well-defined finite value). This lecture presents some basic criteria for convergence. In subsequent lectures, we will develop several more criteria, and also consider several key examples (geometric series, p-series, and power series). The main objective is to understand Taylor series. At the moment, we really only have two example to work with: the harmonic series (and its alternating version 1 − 12 + 13 − 14 + · · · ), and the geometric series 1 + 12 + 14 + · · · (and its alternating version). So we will mainly compare to these for now. Soon we will have more examples to work with. The reading for today consists of three items: Gottlieb §30.5 pp. 969-972 (on the comparison test1 ), the handout “Nth term test, comparison test” (under “reading for the course”), and the solutions to Janet’s handout with the same title. The homework is problem set 16 (which includes weekly problem 15). You should begin working on weekly problems 16 and 17. 2 The N th term test Recall that informally, a series has a well-defined “sum” if there is some number, call it L, such that the partial sums of the series give better and better approximations of L, eventually staying within any chosen margin of error of L. Thinking in these terms, observe the following: for if n is large enough that the partial sums sn remain within some margin of error of L, then the terms of the series must be small enough that they cannot possibly push the partial sum out of this margin. In other words, the terms must remain in some margin of error of 0. This suggests the following criterion. Theorem 2.1 (N th term test). If the series a1 + a2 + a3 + · · · converges, then lim an = 0. n→∞ The contrapositive of this statement is important enough that I will state it separately. Corollary 2.2. If lim 6= 0 (or does not exist at all), then the series a1 + a2 + a3 + · · · diverges. n→∞ Example 2.3. Consider the series 1 + (−1) + 1 + (−1) + · · · . The limit limn→∞(−1)n does not exist, so the N th term test implies that this series diverges. It is very important to note that the converse of theorem 2.1 is not true in general. The classic example of this is the harmonic series. Example 2.4. Consider the harmonic series 1 + 12 + 13 + 14 + · · · . This series passes the N th term test since limn→∞ = 0, but it still diverges. Thus while failing the test guarantees that a series diverges, passing it does not guarantee that it converges. 1 This portion of the text unfortunately sometimes invokes various convergence tests that we have not covered yet, due to the different order of the text. You may disregard these remarks for now if you wish. 1 3 The direct comparison test We begin with the most basic form of the comparison test. This test will be refined somewhat in section 5. Another version, called the asymptotic comparison test, will be presented in a later lecture. Just like with improper integrals, convergence of series can sometimes be established by bounding a series by a series known to converge (and, by contrapositive, divergence of a series can be established by bounding it below by a series known to diverge). P∞ P∞ Theorem 3.1 (Direct comparison test). Suppose that k=1 an and k=1 bn are two series with nonnegative terms such that 0 ≤ an ≤ bn for all n. Then: P∞ P∞ • If k=1 bn converges, then k=1 an converges. P∞ P∞ • If k=1 an diverges, then k=1 bn diverges. Note that I have written these series as starting from k = 1, but this is not essential; this test will work no matter what indexing convention is used. P∞ Example 3.2. Consider the series k=1 √1k . This can be compared to the harmonic series: for all k, √1k ≥ k1 . Since the harmonic series diverges, this series must diverge as well. 1 + · · · . Then since 31n ≤ 21n , we can compare to the geometric Example 3.3. Consider the series 1 + 31 + 19 + 27 1 1 1 series 1 + 2 + 4 + 8 + · · · . Since the latter series converges, this series converges as well. 4 Initial segments do not matter An important observation about series is that their convergence (or divergence) does not depend on any finite number of terms: it only depends on the eventual behavior of the series. This is a bit of a tricky notion to comprehend: no individual term of the series determines whether it converges, yet all terms taken together do determine whether it converges. This can be expressed by the following. P∞ Theorem 4.1. Consider any series n=1 an . Then for any positive value N , this series converges if and P∞ only if the “tail” n=N converges. Therefore to study convergence of a series, is suffices to study convergence of the series after any finite initial segment has been removed. ∞ X 1 . This is the same as the harmonic series, except that an initial Example 4.2. Consider the series n n=1000 segment has been removed. Therefore, since the harmonic series diverges, this series diverges as well. 5 The limit comparison test Bearing in mind that initial segments do not matter, we can refine the direct comparison test to give a test that only uses long-term behavior. The result is the following test. Theorem 5.1 (Limit comparison test). Consider two series ∞ X k=1 ak and ∞ X bk , and suppose that for k k=1 sufficiently large, both series have only positive terms. Suppose also that limk→∞ an bn = L. Then: • If 0 < L < ∞, then either both series converge or both series diverge. P∞ P∞ • If L = 0 and k=1 bk converges, then k=1 ak converges as well. P∞ P∞ • If this limit diverges to +∞ (or, you could say, L = +∞), and k=1 bk diverges, then k=1 ak diverges as well. 2 Here is a rough sketch of the proof of this result. If L is a positive constant, then this means in particular that for sufficiently large k, ak /bk ≤ 2L, so ak ≤ 2Lbk , and also ak /bk ≥ L/2, so that bk ≤ 2Lak . Therefore the comparison test can be used in either direction; the convergence (or divergence) of one series implies the convergence (or divergence) of the other. The 0 and ∞ cases are similar, but only work in one direction, since only one bound (upper or lower) can be obtained. ∞ X n2 + n + 1 . It would be a nightmare to try to set of a comparison Example 5.2. Consider the series 3n3 + 31n + 7 n=27 directly. However, just by looking at this series, is seems like it should be very comparable to the harmonic series. Indeed, taking the ratio and passing to the limit: (n2 + n + 1)/(3n3 + 31n + 7) 1/n n3 + n2 + n lim n→∞ 3n2 + 31n + 7 n3 + n2 + n 3n3 + 31n + 7 1 + 1/n + 1/n2 = lim n→∞ 3 + 31/n2 + 7/n3 1 = . 3 = Therefore since both series have positive terms, the limit comparison test applies: the fate of each series ∞ X n2 + n + 1 . is intertwined with the other. Since the harmonic series diverges, so does the series 3n3 + 31n + 7 n=27 Example 5.3. Consider the series ∞ X k=52 7 . This looks like it could be compared with the familiar 2k − 12 ∞ X k 1 geometric series . Indeed, both series are positive, and the ratios of corresponding terms are 27·2 k −12 , 2k k=1 which goes to 7 as k → ∞.Therefore one converges if and only if the other does; since we know that ∞ X 7 1 + 12 + 14 + · · · converges, then also converges. 2k − 12 k=52 3