By I. Craw

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This is sometime called a nondecreasing sequence. • There is a similar definition of a decreasing sequence. • What does it mean to say that a sequence is eventually increasing? • A sequence which is either always increasing or always decreasing is called a monotone sequence. Note that an “arbitrary” sequence is not monotone (it will usually sometimes increase, and sometimes decrease). Nevertheless, monotone sequences do happen in real life. 1415 . . is how we often describe the decimal expansion of π.

Example. 13. Exercise. Evaluate lim x→0 log(1 + x) . sin x sin x = 1. x Solution. This is spurious because we need the limit to calculate the derivative in the first place, but applying l’Hˆopital certainly gives the result. 14. Example. 3 Rolle and the Mean Value Theorem We can combine our definition of derivative with the Intermediate Value Theorem to give a useful result which is in fact the basis of most elementary applications of the differential calculus. Like the results on continuous functions, it is a global result, and so needs continuity and differentiability on a whole interval.

Show that {an } is 7 + an convergent, and find its limit. Solution. 89 . . and it looks as though the sequence might be increasing. Let f (x) = 6(1 + x) , 7+x so f (an ) = an+1 . 2. BOUNDEDNESS AGAIN 25 By investigating f , we hope to deduce useful information about the sequence {an }. 6 − 6(1 + x) 36 = > 0. (7 + x)2 ((7 + x)2 Recall from elementary calculus that if f (x) > 0, then f is increasing; in other words, if b > a then f (b) > f (a). We thus see that f is increasing. Since a2 > a1 , we have f (a2 ) > f (a1 ); in other words, a3 > a2 .

### Advanced Calculus And Analysis by I. Craw

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