In the early 1700s, mathematics had just taken off like a jet plane, shortly to be more like a rocket. Newton and Leibniz had just invented Calculus, and Johann Bernoulli, a brilliant Swiss mathematician, had turned his mind to inspect everything that the theory could do, and had asked the big theoretical questions that would drive the engine of mathematical discovery in the next hundred years or so. It was an exciting time in which to be a mathematician.
William Dunham, of Muhlenberg University has made a particular study of Leonhard Euler, one of the greatest mathematicians of all time, born in Basel, and a protegee of the famous Bernoulli. In his book "The Master of Us All", Dunham describes some of the most accessible and interesting of the discoveries of Euler, making a point to underscore the intuitive methods Euler used, which though not acceptable as formal proof today, reveal the amazing mind of Euler.
To the layman, the idea of an infinite series might be a little strange. An infinite series is simply an addition of an infinite number of terms. Generally, a sum of infinitely many terms will produce an infinite number. However, if the numbers are very small, the sum could be finite. Such finite-valued "infinite sums" can be highly useful; many useful numbers can be approximated as "truncated"infinite series (infinite sums of which only a finite number of terms have been added; what is omitted is carefully calculated to be less than an acceptable error tolerance).
Consider the infinite sum 1/2 + 1/4 + 1/8 + 1/16 + .... With a little thought, it can be seen that this sum can never be greater than 1. As Zeno observed, if you walk half a mile, then walk half of the remaining half mile, and then half of that, and keep doing this, you'll never walk more than a mile. In the chart below, the blue bars represent the terms that we want to add up, and the pink bars represent the running subtotals. As you can see, the subtotals approach 1, and rise no higher.
In contrast, the series 1 + 1/2 + 1/3 + 1/4 + ... can be seen to be infinite. Observe the sums climbing steadily. (They do slow down, but not enough to make the sums approach a finite value.)
It is harder to show, but the series 1 + 1/4 + 1/9 + 1/16 + ... , (the sum of the reciprocals of the square numbers, 1, 4, 9, 16, 25, ...) is a finite sum. (See below. The sum is about 1.64.) The story of how Euler showed that it adds up to the unlikely value of (Pi)^2/6 (Pi squared divided by 6) is entertainingly recounted by Dunham in his wonderful book. This derivation is characteristically Eulerian!
The discovery above is topped by a result that connects an infinite series consisting of the reciprocals of the squares of the positive integers on the left, with an infinite product of factors of the form (1 - 1/P^2) on the right:
Bill Dunham's book, beautifully written, combines stylish exposition with absolutely fascinating content. Euler, a contemporary of J.S.Bach, showed a similar brilliance and creativity to the musical genius, and both had similarly enormous influence over the development of their respective areas. Both had many children: Bach had 20, and Euler had 13, and in each case, only a handful lived to adulthood. Finally, both were afflicted with diseases of the eye. While Bach died at the age of 65, Euler lived to be nearly 80.
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