Diophantische Approximationen by J. F. Koksma (auth.)

By J. F. Koksma (auth.)

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Again, contrary to intuition, both series (opposite page) are divergent. Yes, the sums of both the series tend to infinity. How many terms of the Harmonic Series do you need to remove before you get a convergent series? This is a problem that continues to puzzle mathematicians. " As it turns out, there are numerous infinite series which do sum to constants. "The Geometric Series", an infinite series of some of the reciprocals, sum to a small number, 2. It is amazing how minor differences in the series can sometimes result in dramatic differences in their sums — 2 for "the Geometric Series", and "tending to infinity" for "the Harmonic Series" and their odd and even subsets.

One of his many mathematical contributions is the expression of K as an infinite product involving only the integer 2, in an infinite nesting of square roots. Viete's expression marked a milestone in the history of mathematics. It was the first equation incorporating the concept of an infinite process, even though it was not explicitly spelt out as such. ) in the equation denotes continuing the process indefinitely. If we think that the expression looks difficult, remember Viete published it in 1593, more than four hundred years ago!

71-series 31 More Euler Series J_ _I_ J_ J_ J_ ^T "T~ l 2 r\ ^T~ ~T~ 3 2 I 5 /-) ~~T~ ^ 2 "T" 7 9 1 82 1 102 1_ \ 1 n 8 1 1 >2+42 1 62 ,r 2 24 J 1_ J 1_ ,r 2 12 See Proof 42 -^ (page 216) ground-breaking achievement with \ , similar beautiful gems rained from the sky with relative ease. By simple arithmetical rearrangements of the terms in I T (the series is absolutely convergent, so we can rearrange the terms without erroneous results — see "A Note of Caution"; page 175), three more Euler series can be obtained, all three as beautiful as the original.

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