Section.5 Absolute and Conditional Convergence Another special type of series that we will consider is an alternating series. A series is alternating if the sign of the terms alternates between positive and negative, such as the alternating harmonic series ( ) n n = + 3 4 + (Recall that the harmonic series is n = + + 3 + 4 + ). Alternating series have a special form; since the sign alternates from term to term, any alternating series can be written as ( ) n a n ; the ( ) n takes care of the alternating sign. Because of the special form of an alternating series, there is an simple way to determine that many such series converge: Theorem 0.0.. The Alternating Series Theorem The alternating series ( ) n a n = a a + a 3 a 4 + converges if all three of the following conditions are satisfied:. Each a i is positive. a n a n+ for all n N, for some integer N 3. a n = 0. In a sense, this test is similar to the nth term test, as it involves taking the it of the terms. However, the Alternating Series Theorem only applies to alternating series while the nth term test can be applied to any series, including alternating series. Note also that the theorem can only tell us that a series converges, but can t tell us that a series diverges. In addition, some alternating series that converge might not satisfy the conditions of the theorem. However, if the alternating series ( ) n a n fails condition 3 of the Alternating Series Test, i.e. then a n 0, ( )n a n 0, which means that the series itself will fail the nth term test and diverges. Examples: Show that the alternating harmonic series converges.
Section.5 The alternating harmonic series, given by ( ) n n = + 3 4 +, has a n = n. So each a i > 0, and it is easy to see that a n a n+ for all n since n > n+. Finally, since n = 0, we see that the alternating harmonic series satisfies the three conditions of the theorem, thus converges. Absolute and Conditional Convergence Definition 0.0.. A series a n converges absolutely if the series of absolute values a n converges. A series converges conditionally if a n converges but a n does not converge. We have already seen that the alternating harmonic series ( ) n n converges, but the harmonic series ( )n n = n does not. So the alternating harmonic series converges conditionally. The series ( ) n ( ) n 3 converges absolutely since the series of absolute values is geometric with r = 3 <. ( n 3) ( )n = ( ) n = 3 It turns out that any series that is absolutely convergent must be convergent itself: Theorem 0.0.3. The Absolute Convergence Test If a n converges, then a n does as well.
Section.5 In other words, if the new series we get from a n by taking absolute values of all the terms is a convergent series, then the original series converges as well. There are several important things to note here: first of all, the theorem is helpful since we already have several tests for series that have positive terms. To test a series that doesn t have all positive terms, we can take absolute values and use one of the previously studied convergence tests (such as the root test or the ratio test) that apply to series with positive terms; if the series of absolute values converges, then the original series does too. If the series is not absolutely convergent, it may still be conditionally convergent, or it may diverge after all. To check, try the Alternating Series Theorem (if the original series was alternating) or the nth term test. Examples Determine if each of the following series converges absolutely, conditionally, or diverges. ( ) n tan( n ) To determine if the series converges absolutely, we need to determine if tan( ( )n n ) = tan( n ) = tan( n ) converges or diverges. Let s try the it comparison test. For large n, n 0 so that tan( n ) 0. It seems that the function tan( n ) behaves similarly to n, so we will replace the numerator with n. We compare tan( n ) to n = : n 3 tan( n ) n 3 n 3 tan( n ) n tan( n ) LR = tan( n ) n sec ( n ) sec ( n ) = sec 0 =. The comparison series converges since it is a p-series with p = 3 > ; since the it of n3 3
Section.5 the quotient of the terms is the it comparison test tells us that tan( n ) converges as well. Since the series of absolute values converges (that is the original series converges absolutely), we conclude that ( ) n tan( n ) converges as well. ( ) n It is easy to see that the series does not converge absolutely by the nth term test: n n ( )n n =. n n n n n n n n Since the it of the terms is non-zero, the series of absolute values diverges. The original series may converge conditionally, or it may diverge; however the result of the previous test indicate that we should try the nth term test on the original terms. If n is even, we already know that If n is odd, then ( )n n =. ( )n n =. 4
Section.5 This tells us that the it of the terms does not exist; so the series fails the nth term test, thus diverges. ( ) n ln n n Let s start by testing for absolute convergence; we need to look at the series ( )n ln n n = ln n n = n ln n. We can definitely integrate test. n ln n by using the u-substitution u = ln n, so we try the integral n ln n dn = u du = ln u + C = ln(ln n) + C. So b dn n ln n b n ln n dn b (ln(ln n) b ) b (ln(ln b) ln(l)) =. Since the improper integral diverges, the series of absolute values does too. Thus the original series is not absolutely convergent, but might still be conditionally convergent. To check, let s try using the Alternating Series Theorem. We have a n = ln n = n n ln n ; each term a n is nonnegative, and it is clear that (n+) ln(n+) < n ln n since (n + ) ln(n + ) > n ln n. Finally, we check Since the it is 0, the original series n ln n = 0. ( ) n ln n n converges. Since it does not converge absolutely, we specify that the series converges conditionally. 5