An Introduction to Mathematics: For Students of Economics by J. Parry Lewis (auth.)

By J. Parry Lewis (auth.)

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But for how long? The answer is that this process will go on for ever, provided that there is always some underemployed labour available, and unless, of course, some other event intervenes. But the fact that it will go on for ever does not mean that income will grow to indefinitely high levels. We may notice that the annual increases in output are 0·1, 0·09, 0·081, 0·0729, ... P. whose first term is 0·1 and whose common ratio is 0·9. 46 ALGEBRA By the sixth month, four of these terms have arisen, each representing an increase over the previous month.

Meanwhile we may obtain some introduction to the answer, as well as to the problems of convergency, if we examine a few geometric progressions. Consider first of all the series 1+2+4+8+16+32+ ... Clearly, by taking enough terms, we may make this sum as large as we like. This is an example of a divergent series. We shall define these concepts shortly, meanwhile let us note that if we wish to make the sum of this series greater than 250,000,000 all that we have to do is to take the first twenty-nine terms, when we shall obtain the sum 268,435,455.

S. = -16 + 8 + 30 + 6 = 28 This shows that the change of sign, and therefore the root, occurs somewhere between -5 and - 2. Proceeding in this way, we will soon find that in fact the root lies between - 3 and - 4. We now have to decide whether to try to find this root more exactly, or to see whether there is a simpler root which we can find first. Substituting integers in the above expression is quite easy: but if we are to find this particular root by this method we will have to substitute fractions, which become complicated.

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