Math 1 D , Week 2 – Cauchy Sequences , Limits Superior and Inferior , and Series

These are the lecture notes from week 2 of Ma1d, the Caltech mathematics course on sequences and series. 1. Limits Superior and Inferior So: most of the definitions and theorems we’ve developed so far for sequences are centered around the concept of convergence – we have lots of ways of talking about when things converge, where they converge to, and under what conditions they will be forced to converge. However, when we’re confronted with a divergent sequence, it is sometimes useful to be able to say more about it than just “it doesn’t converge!” For example, the sequences 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, . . . 1 3 , 2 3 , 1 4 , 3 4 , 1 5 , 4 5 , 1 6 , 5 6 , 1 7 , 6 7 , . . . both diverge, and yet both exhibit very clear behaviors at infinity – specifically, both sequences seem to “tend” to both 0 and 1 at infinity. The following definition helps us offer a canonical way of talking about such limiting behaviors at infinity, even when looking at such divergent sequences: Definition 1.1. For a sequence {an}, set xn = sup{am : m ≥ n}. We then define the limit superior of an as lim sup n→∞ an = lim n→∞ xn. Similarly, if we set yn = inf{am : m ≥ n}, we can then define the limit inferior of an as lim inf n→∞ an = lim n→∞ yn. Example 1.2. If {an} is the sequence given by 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, . . . then for any n we have that sup{am : m ≥ n} = 1, inf{am : m ≥ n} = 0,