Sequences Series and Limits

Studying sequences and series is the best way to gain intuition about the rather perplexing notions of arbitrarily large numbers (infinity) and inrtnitesimally small (but nonzero) numbers. We gain such understanding by using the idea of the limit of a sequence of numbers. Thus, from a mathematical perspective, this chapter provides very useful background to the important property of continuity of a function, which we will explore fully in chapter4. There are also some interesting economic applications of series and sequences, in particular the notion of discounting a future stream of payments or receipts, which is a critical aspect of judging the value of an investment by a business or a government.

A sequence is simply a succession of numbers. For example, the sequence of numbers 1, 4,9. 16, ... appears lo consist of the squares of the natural numbers

(i.e.. I2, 22. 32. 42____). It is common to see questions on IQ or mathematical aptitude tests asking one to (ill in the next number in a sequence, which involves figuring out or guessing the formula that generates the numbers of the given terms of the sequence. We could write that formula for the sequence above as

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