 Some of the greatest work of Leonard Euler centered around series summation. In general, his methods are not considered rigorous today. However, in most cases, a rigorous proof can be found confirming Euler's results. Roughly speaking, these rigorous proofs rely on the theorem that if a series is absolutely convergent, then essentially we can treat it like a polynomial of infinite degree. After Euler found the sum of the reciprocals of the squares, Euler considered a generalization, the sum of the reciprocals of the n-th powers, in various observations on infinite series. This series defines what is now called the Riemann-Zeta function. Euler's main result was showing that this series could be written in product form, where the factors have the form p to the n over p to the n-1, where p is a prime number. And this connects the Riemann-Zeta function to the theory of prime numbers. To prove this, Euler assumed the series had some x. Now if we multiply both sides by 1 over 2 to the n, then we get, and now we can subtract to get, and the important thing here is that our subtraction has eliminated all terms 1 over a to the n, where a has a factor of 2. And again, remember we're treating this as an absolutely convergent series so we're allowed to do this. But now, if we multiply it by 1 over 3 to the n and subtract, we can eliminate all terms 1 over a to the n, where a has a factor of 3. And lather rins repeat, and we find that x times this product of factors that look like 1 minus 1 over prime to the n is equal to 1, and so we can solve for x. And since x was the sum of the series, then the sum of the series of reciprocals could be expressed in product form. Euler then noted that if n equals 2, then from his previous result, that pi squared over 6 is the sum of the reciprocal squares, well that's really this formula where n is equal to 2, and so we get. But note that all the denominators are differences of squares, so using our factorization for the differences of squares, we find, and so pi squared over 6 can be expressed as where the numerator consists of two factors of each prime, and the denominator consists of numbers 1 more and 1 less than each prime. And we could go further. Since Euler had found expressions for the sum of the reciprocal 4th, 6th, 8th, 10th, and 12th powers, these could be turned into other product forms. For example, from the sum of the reciprocal 4th powers, Euler found a product form, and if we divide this by the expression for pi squared over 6, we obtain. Euler didn't give the details but we can fill them in from the product form of the sum of the reciprocal 4th powers we have, and consequently, well you can do your own homework from this point.