1 The Bridge Between Continuous and Discrete

In the early 1730s, Leonhard Euler (1707–1783) astonished his contemporaries by solving one of the most burning mathematical puzzles of his era: to find the exact sum of the infinite series 11 + 1 4 + 1 9 + 1 16 + 1 25 + · · · , whose terms are the reciprocal squares of the natural numbers. This dramatic success began his rise to dominance over much of eighteenth-century mathematics. In the process of solving this then famous problem, Euler invented a formula that simultaneously completed another great quest: the two-thousand-year search for closed expressions for sums of numerical powers. We shall see how Euler’s success with both these problems created a bridge connecting continuous and discrete summations. Sums for geometric series, such as 11 + 1 2 + 1 4 + 1 8 + · · · = 2, had been known since antiquity. But mathematicians of the late seventeenth century were captivated by the computation of the sum of a series with a completely different type of pattern to its terms, one that was far from geometric. In the late 1660s and early 1670s, Isaac Newton (1642–1727) and James Gregory (1638–1675) each deduced the power series for the arctangent, arctan t = t− t3 3 + t 5 5 − · · · , which produces, when evaluated at t = 1, the sum π4 for the alternating series of reciprocal odd numbers 1− 13 + 15 − 17 + 19 − · · · [133, pp. 492–494], [135, pp. 436–439]. And in 1674, Gottfried Wilhelm Leibniz (1646– 1716), one of the creators of the differential and integral calculus, used his new calculus of infinitesimal differentials and their summation (what we now call integration) to obtain the same value, π4 , for this sum by analyzing the quadrature, i.e., the area, of a quarter of a unit circle [133, pp. 524–527]. Leibniz and the Bernoulli brothers Jakob (1654–1705) and Johann (1667– 1748), from Basel, were tantalized by this utterly unexpected connection