Zeno's Dichotomy paradox rests on the inability to conceive of adding infinitely many numbers to arrive at a finite sum. However, a glance at the diagram that accompanies the description of the paradox (note 2) shows quite clearly that
1/2 + 1/4 + 1/8 + 1/16 + ... = 1
How can we justify such a statement?
The expression on the left side of the equation is called an infinite
series. Any infinite sum of numbers is an infinite series. The question
we are considering is: under what conditions, if any, can an infinite series
be said to have a finite sum? More importantly, what might we mean
when we perform any operation an infinite number of times? This is
the first substantial attempt in Western thought to wrestle with the notion
Modern mathematics deals with this problem as follows. Given an infinite series, like 1/2 + 1/4 + 1/8 + 1/16 + ... , we consider the corresponding infinite sequence of partial sums
1/2, 1/2 + 1/4, 1/2 + 1/4 + 1/8, 1/2 + 1/4 + 1/8 + 1/16, ...
which here is equivalent to 1/2, 3/4, 7/8, 15/16, ... . The sequence of partial sums is formed by successively adding on the next term of the original series. If there is some number S with the property that the terms of this sequence of partial sums get closer and closer to S the farther out in the sequence you go, then we say that the infinite series converges to S, that is, that S represents the sum of the series. In our example, the sequence of partial sums can be rewritten in the form
1 - 1/2, 1 - 1/4, 1 - 1/8, 1 - 1/16, ...
from which it is clear that the
terms get closer and closer to 1 as one goes farther out in the sequence.
So we can take S = 1 in this example and assert that the series
does sum to 1.
Of course, not all infinite series converge to a sum. Simple examples like
1 + 2 + 3 + ... or even 1 + 1 + 1 + ...
should be enough to convince us
of this. The question of how one can tell whether a series converges or
not is in general a difficult one, and we shall be content to consider
for the moment just one kind of series for which the answer to this question
Infinite series are not as strange and obscure as they may seem. For instance, every number that is representable as a repeating decimal can be interpreted as an infinite series. Take the familiar number .3333... . Expanding it in terms of powers of 10 gives us
The number .513513513... has the representation
These series all share the same property: each term of the series can be determined by multiplying the previous term by the same constant value. For instance, in
each term is 1/2 times the previous term; in the series for .3333..., each term is 1/10 times the previous term; and in the series for .513513513..., each term is 1/1000 times the previous term. Series of this type are called geometric series. In general, a geometric series has the form
where a represents the initial
term and r the common factor that generates each successive term.
To determine the conditions that allow a geometric series to converge, we must consider the sequence of partial sums. In preparation for this analysis, let us review some algebra.
We recall that certain polynomials can be factored; in particular,
This suggests a pattern, which can be verified by multiplying out the factors on the right:
In general, this pattern takes the form
It is more useful to us however to write it as
For now we can replace x with r and multiply by a to get a formula for the partial sum of a geometric series:
We can return to our analysis now. The sequence of partial sums of our geometric series is
which by the formula we just derived can be rewritten as
The further we go out into this sequence, the larger the power of r in the numerators of these fractions. So as long as r is a number between -1 and 1, we can be sure that higher and higher powers of it will get smaller and smaller. Consequently, the terms of this sequence are getting closer and closer to
It follows that the geometric series must converge to this value. In other words, we have determined the sum of a geometric series in general:
This result verifies the calculations we made earlier. The series from the Dichotomy paradox corresponds to a = 1/2 and r = 1/2, and this formula tells us that the sum is (1/2)/(1-1/2) = 1 as we knew already. The series for the decimal number .3333... has a = 3/10 and r = 1/10, and the formula gives the sum
(3/10) / (1 - 1/10) = (3/10) / (9/10) = 3/9 = 1/3 .
Finally, the series for .513513513... has a = 513/1000 and r = 1/1000, and the formula gives the value
(513/1000) / (1 - 1/1000) = (513/1000) / (999/1000) = 513/999 = 19/37 ;
this gives us the fractional equivalent
of the decimal: .513513513... = 19/37 .
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