Sequences (12.1) A sequence is a list of numbers. More technically, a sequence is a function with domain the positive integers. We could write f (n) for the n-th term (number) of the sequence, but more usually we write something like an for the n-th term. Sometimes, people write {a1 , a2 , a3 , . . . } to indicate a sequence, or {an } . Other people prefer (an ) . n=1 n=1 Examples 1. an = 2. an = (-1)n 1 1 1 . So a1 = - , a2 = , a3 = - , and so on. 3n 3 9 27 n2 n 1 2 10 . So a1 = , a2 = , and a10 = . +1 2 5 101
3. Let pn be the n-th prime number. So p1 = 2 (the first prime number by custom is 2 not 1), p2 = 3, p3 = 5 and p4 = 7. Euclid proved that there are infinitely many primes, so this makes sense. But no one knows a formula for the n-th prime. 4. Let f1 = 1, f2 = 2 and let fn = fn-1 + fn-2 for n > 2. So f3 = f2 + f1 = 2 + 1 = 3 and f4 = f3 + f2 = 3 + 2 = 5. These are the Fibonacci numbers. They are defined "recursively": the n-th term is defined in terms of the preceeding terms. There is a formula for them, however: 1 fn = 5 1+ 5 2
n
-
1- 5 2
n
.
2n . If n is large, we find that an is close in n+3 200 value to 2. For example, if n = 100 then an = a100 = = 1.94 , but if n = 1000, 103 2000 an = a1000 = = 1.994 . We say an converges to 2, or that the limit as n goes to 1003 infinity of an is 2. We write an 2 or lim an = 2. Limits of sequences Consider an =
n
There is a technical definition: We say an L if for every > 0 there is a number n0 such that if n > n0 then |an - L| < . This means, we can make an as close as we want to L, by making n sufficiently large. So controls how close an is to L, and n0 controls how large n needs to be. Examples 1. lim 2n 2x = 2. this Actually, is the same as the limit lim = 2 and can be n n + 3 x x + 3 found in the same way. 1
2. an = 0.
n2
n n x . This converges: lim 2 = 0. This is the same limit as lim 2 = n n + 1 x x + 1 +1 3 4
n
3n 3. an = n . This is an = 4 4. an =
0. In general, if 0 < c < 1 then lim cn = 0.
n
n3 x3 . The limit of this is the same as the limit lim x , which can be found x 2 2n (using L'Hospital's rule three times). The limit is 0. More generally, if c > 1 then nk lim n = 0. n c
5. an = (-1)n . This is the sequence {-1, 1, -1, 1, -1, . . . }. It has no limit (it does not converge).
1 6. an = 1 + n
. Try n = 100. Get a100 = (1.01)100 = 2.704 . The limit turns out to x 1 be e = 2.71828 . This is the same limit as lim 1 + = e, which can be found x x by taking logarithms and using L'Hospital's rule.
n n
n
Limit theorems. Suppose lim an = a and lim bn = b. Then 1. lim (an + bn ) = a + b.
n n n
2. lim (an bn ) = ab. 3. lim (an /bn ) = a/b provided b = 0. We say a sequence an is increasing if an+1 an for all n. We say it decreases if an+1 an for all n. A sequence is called monotone if it is increasing, or if it is decreasing. An important theorem says that if a sequence is increasing and bounded above (so there is a number M such that an < M for all n), then the sequence is convergent. Likewise, if a sequence is decreasing and bounded below, it must converge. An example: Consider the sequence a1 = 0.4, a2 = 0.44, a3 = 0.444, a4 = 0.4444, etc. It is increasing, and it is bounded above (by 1 or by 0.5, for example). So it must converge. Notice that 10an+1 - an = 4. If we call the limit a, we have can take the limit of that equation to get the equation 10a - a = 4, so a = 4/9.
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