By B. Thomson, et al.
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Extra info for Elementary Real Analysis
Or we could note that the sequence starts with 2 and then all the rest of the terms are obtained by adding 2 to the previous term: “Consider the sequence whose ﬁrst term is 2 and whose nth term is 2 added to the (n − 1)st term,” that is, xn = 2 + xn−1 . Often an explicit formula is best. Frequently though, a formula relating the nth term to some preceding term is preferable. Such formulas are called recursion formulas and would usually be more eﬃcient if a computer is used to generate the terms.
There is another aspect of structure of the real number system that plays an equally important role as the algebraic structure and that is the order structure. Does the limit operation preserve that order structure the same way that it preserves the algebraic structure? For example, if s n ≤ tn for all n, can we conclude that lim sn ≤ lim tn ? n→∞ n→∞ In this section we solve this problem and several others related to the order structure. These results, too, will prove to be most useful in handling limits.
Every arithmetic progression could be given by a formula xn = c + (n − 1)d or a recursion formula x1 = c xn = xn−1 + d. Note that the explicit formula is of the form xn = f (n), where f is a linear function, f (x) = dx + b for some b. 1 shows the points of an arithmetic progression plotted on the line. If, instead, you plot the points (n, xn ) you will ﬁnd that they all lie on a straight line with slope d. 2. 2. A geometric progression. A variant on the arithmetic progression is obtained by replacing the addition of a ﬁxed amount by the multiplication by a ﬁxed amount.