# How Euler Did It If the first partial sum of this series is 1 and the second is 3

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2 Now, Euler integrates both sides. On the left, he gets desired nth partial sum of the harmonic series. On the right, he gets n x x x x integrals had not yet evolved, so instead of writing Euler adds that we should take it = 0 if x = 0, and then set x 0 x Taking x = 1 gives the n n 1 x dx. Our modern notation for definite 1 x n 1 x dx 1 x = 1. So Euler uses this integral as the value for the nth partial sum of the harmonic series, even if n is not a whole number. For most values of n, the integral too difficult to find that n 11 x dx 0 1 x 1 x 1 dx = dx = 2 2ln2 1 x 1+ x is rather difficult to evaluate. In the case 1 1 0, thus giving a value to the n = 2, though, it is not th partial sum. Euler can always find more to do with a new technique. This integration idea works because integration gives a relation between geometric and harmonic series. In particular, (overlooking, as Euler did, the constant of integration) 1 2 n 1 dx= ( 1 + x+ x x ) dx= x x 2 3 n n 2 n x x x x If we integrate again, then on the right hand side we get n+ 1 x x x x n + 1. ( n ) Here, the denominators are products of consecutive integers. These are tantalizingly like, but are not exactly squares. The Basel Problem seems to be lurking nearby. Euler, though, sees a way to get the perfect squares in the denominators that he wants so badly. He divides by x before integrating again. He sees that 2 n 2 3 n 1 x x x x x x x dx= x x 2 3 n nn, again, neglecting the constant of integration. Euler doesn t have a very good notation for double integrals, either, so he writes this in a way that we would now regard as nonsense: n dx 1 x dx = x 1 x 4 9 n Euler does give us instructions about how to evaluate this, and they translate into modern notation to be 1 n 1 x1 y dy dx x 1 y 0 0 To solve the Basel Problem all Euler has to do is evaluate this integral and then take its limit as n goes to infinity. This does not seem like progress, because this integral is very difficult to evaluate. However, Euler uses a very clever approximation technique to get a numerical value, , for this integral. The series itself converges rather slowly, and you would have to sum more than 30,000 terms to get this degree of accuracy directly. This value does not mean much to most people, but among his other skills, Euler is a fantastic calculator. He sees this value is close to 2 π 6 functions, in particular, the Taylor Series for sin x x. Armed with this hint, he begins to attack the Basel Problem using series for trigonometric years later, he publishes his solution and the world of mathematics knows his name.. In fact, Euler was the first one to call them Taylor s series. Five [D] Dunham, William, Euler The Master of Us All, MAA, Washington, D. C., 1999.

3 Ed Sandifer is Professor of Mathematics at Western Connecticut State University in Danbury, CT. He is an avid marathon runner, with 31 Boston Marathons on his shoes, and he is Secretary of The Euler Society ( How Euler Did It is updated each month. Copyright 2003 Ed Sandifer

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