18.01 Calculus Jason Starr Fall 2005 Lecture 30. December 6, 2005 Practice Problems. Course Reader: 6C-2. 1. Sequences By definition, a sequence of real numbers is a rule assigning to each counting number n an associated real number an. The integer n is called the index of the sequence. Usually the index begins with n = 1, but occasionally it begins with another integer (sometimes 0). Sequences are often specified by giving the first few values, and letting the reader infer the rule, e.g., 1 1 1 a1 = , a2 = , a3 = , . . . 1 2 3 It is always better to give a precise definition of each sequence, e.g., 1 an = , n = 1, 2, . . . n The most common notation for a sequence is (an)n≥1.18.01 Calculus Jason Starr Fall 2005 A sequence (an)n≥1 converges to a limit L if the sequence becomes arbitrarily close to L, and stays arbitrarily close to L. More precisely, the sequence converges to L if for every positive number �, there exists an integer N (depending on the sequence and �) such that for every integer n ≥ N, an − L < �.| | In other words, the tail of the sequence aN , aN +1, aN +2, . . . are all numbers in the interval (L −�, L + �). A sequence cannot have more than 1 limit: given 2 potential limits L1 and L2, simply take � = L1 − L2/2 in the definition above. A sequence which has a limit is said to converge, and | |the limit is denoted by, L = lim an. n→∞ A sequence which does not have a limit is said to diverge. Examples. (i) Let L be a fixed real number. The sequence an = L, n = 1, 2, . . . converges to L. (ii) The sequence an = n diverges. In a precise sense, this sequence “diverges to ∞”. (iii) The sequence an = (−1)n diverges, even though it is bounded (it never gets bigger than 1 or smaller than −1). (iv) Let r be a real number. The sequence an = rn, n = 0, 1, 2, . . . converges to 0 if |r| < 1 and diverges if |r| > 1. There are 2 remaining cases. If r = −1, then an = (−1)n diverges. If r = 1, then an = 1 converges to 1. 2. Tests for convergence/divergence. One useful test for convergence is the Squeezing Lemma. The squeezing lemma. Let (an)n≥1, (bn)n≥1 and (cn)n≥1 be sequences such that for every index n, an ≤ bn ≤ cn. In other words, the sequence (bn) is “squeezed” between the sequences (an) and (cn). If (an) and (cn) converge, and if, lim an = lim cn, n→∞ n→∞ then also (bn) converges and its limit equals the limit of the other 2 sequences. Another test for convergence is the Monotone Convergence Test. A sequence (an)n≥1 is called non-decreasing if for every index n, an+1 ≥ an. Similarly, a sequence (an) is non-increasing if for every index n, an+2 ≤ an. A sequence which is either non-decreasing or non-increasing (but not both increasing and decreasing) is called monotone. A sequence (an) is bounded above if there exists a real number u such that for every index n, an ≤ u. The number u is an upper bound for the sequence. A sequence (an) is bounded below if there exists a real number l such that for every index n, an ≥ l. The number l is a lower bound for the sequence.� � � � � � 18.01 Calculus Jason Starr Fall 2005 Monotone Convergence Test. A non-decreasing sequence converges if and only if it is bounded above. In this case, the limit of the sequence is the least upper bound for the sequence. Similarly, a non-increasing sequence converges if and only if it is bounded below and the limit is the greatest lower bound for the sequence. 3. Series. Given a sequence (an)n≥1, there are 2 important related sequences. The first is the sequence of partial sums, (bn)n≥1, defined by, nbn + an = ak .= a1 + a2 + ···k=1 The second is the sequence of partial absolute sums, (Bn)n≥1, defined by, nBn = |a1+ a2+ |an= ak |.| | | + ··· || k=1 If the sequence of partial sums (bn)n≥1 converges, the limit is called the series of (an)n≥1, and is denoted by, n∞ak := lim bn = lim ak . k=1 n→∞ n→∞ k=1 In this case is is said the series �k ak converges. If the sequence of partial absolute sums (Bn)n≥1 converges, it is said the series converges absolutely. Although it is not obvious, if the k ak series converges absolutely, then the series converges (this is a basic theorem from course 18.100). If a series converges but does not converge absolutely, sometimes it is said the series converges conditionally. Examples. The harmonic sequence is the sequence an = 1/n. As will be shown soon, the harmonic series 1/n diverges to ∞. The alternating harmonic sequence is,n (−1)n an = . n The alternating harmonic series, ∞� (−1)n , n n=1 does converge. This will also be shown soon. Since the sequence of partial absolute sums for the alternating s equence equals the sequence of partial sums for the harmonic sequence, the alternating harmonic series does not converge absolutely. It only converges conditionally. As counter-intuitive as this might sound, the terms in the alternating harmonic series can be rearranged so that the sum converges to any real number you like! This sounds ridiculous: finite sums are independent of the order in which the summands are added, so how could this fail for� � � �� 18.01 Calculus Jason Starr Fall 2005 infinite sums? The answer is quite simple. Because the harmonic series 1/n diverges, the same � n is true for 1/2n . Thus, add it up a very large number of only the (positive) even terms in the alternating harmonic series to make the partial sum bigger than, say, 106 . Now add only the first odd term −1/2. This has a negligible effect. Now add a large number of the remaining even terms to make the partial sum bigger than 107 . Now add one more odd term, −1/3. Continuing in this way, eventually every term in the sequence contributes to one of the partial sums. But because we add positive terms with a much higher frequency than negative terms, the sequence of partial sums is diverging to +∞. Similarly, we could negative terms with a very high frequency and make the partial sums diverge to −∞. Now it is not so surprising that by adding the terms in a c areful order, we can make the partial sums converge to any value we like. The pathology of the preceding paragraph occurs with any conditionally convergent series. It is a very important fact that every absolutely convergent series has
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