The same is true for any other values of less than or equal to 1: the sum diverges.Īs it stands the Euler zeta function S(x) is defined for real numbers x that are greater than 1. So you recover our original sum, which, as we know, diverges. But what happens when you plug in a value of that is less than 1? For example, what if you plug in ? Let’s see. is what’s called a function, and it’s called the Euler zeta function after the prolific 18th century mathematician Leonhard Euler. For every, the expression has a well-defined, finite value. Now what happens when instead of raising those natural numbers in the denominator to the power of 2, you raise it to some other power ? It turns out that the corresponding sumĬonverges to a finite value as long as the power is a number greater than. Then the results you get get arbitrarily close, without ever exceeding, the number Mathematicians say the sum converges to, or more loosely, that it equals If you take the sequence of partial sums as we did above, You might recognise this as the sum you get when you take each natural number, square it, and then take the reciprocal: To understand what that is, first consider the
He had been working on what is called theĮuler zeta function. But Ramanujan knew what he was doing and had a reason for In the work of the famous Indian mathematician Srinivasa Ramanujan in 1913 So where does the -1/12 come from? The wrong result
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