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By Saul A. Basri

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1 What the Greeks had refused to accept was the fact that an infinite sum may add up to a finite value, that is, may converge to a limit. Before we can take a closer look at infinite series, we must explain what is meant by the limit of an infinite sequence, or progression. A sequence is simply a row of numbers, written as a 10 a 2, a 3, • . • , an, . . , where there is usually (though not always) some rule that tells us how to obtain the next number in the sequence. -that the sequence will never come to an end.

6 . 6 ... )1(1 . 3 . 3 . 5 . 5 . 7 ... ) to 7T12, both of which are irrational numbers and cannot therefore be written as fractions. Thus, the rational numbers are closed under the four basic arithmetic operations only so long as we apply these operations a finite number of times. When we apply them infinitely many times, the result may transcend the realm of the rationals. Today the existence of irrational numbers disturbs no one anymore. ) are irrational, as are the numbers 7T and e and combinations of them.

But this is not so. On the other hand, the series consisting of the reciprocals of all twin primes-pairs of primes of the form p and p + 2 (such as 3 and 5, 5 and 7, 11 and 13, and so on}-does indeed converge. But a word of warning must be added here: we do not know whether these twin primes are finite or infinite in number. Most mathematicians believe that their number is infinite (even though they are distributed even more sparsely among the natural numbers than the primes themselves); but until we know for sureand this may not be very soon-we are not entirely justified in regarding this series as an infinite one.

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