 Euler's first publication was in 1725. He died in 1783, having published over 500 books and papers. After his death, his publication rate slowed, and he only published about 300 books and papers after his death. Part of the reason Euler could write almost 900 books and papers is that he returned to the same problem multiple times with different approaches. So after having shown that the sum of the reciprocal squares is pi squared over six using the relationship from the factorization of the McLaren series, Euler presented a different approach in a 1735 paper. And this emerges as follows. Euler noted that the series expansion failed if y equaled zero. But in that case we'd have where x is a solution to sine x equal to zero. Now one solution is x equals zero. Since x equals zero is one solution, we can remove a factor of x to get. And so the remaining solutions are going to be from setting the second factor equal to zero. And as before, Euler assumed that this could factor as... Now the solutions to sine x equals zero besides x equals zero are plus or minus pi, plus or minus 2 pi, plus or minus 4 pi, and so on. So our factorization will be... And because we have factors that are the sum and difference of the same terms, we can multiply them pairwise to get. And consequently this infinite series can be expressed as... Now the x squared term in the expansion comes from the sum of the x squared terms in the factors. And so we have... Which gives us a different way of deriving this summation. But wait, there's more. To extend this result, Euler used some well-known, at-the-time formulas. So again, let alpha be the sum of terms and let beta be the sum of the pairwise products, then the sum of the squares of the terms will be alpha squared minus 2 beta. So from this sum, we can find the sum of the squares if we knew the value of the sum of the pairwise products. But the sum of the pairwise products is how we would determine the x to the fourth coefficient in the expansion. And since we have... We know that the coefficient of the x to the fourth term is the sum of the pairwise products. And so our formulas give us... And so we can conclude... Similarly, if we have our sum, sum of the pairwise products, and sum of the triple products, then the sum of the cubes can be expressed as... And we already know the sum, the sum of the pairwise products. And the sum of the triple products will give us the x to the sixth coefficient in the expansion. And so we have gamma equal to 1 7 factorials. And so we find... And in this way, Euler found the sum of the reciprocal fourth powers, the sum of the reciprocal sixth powers, the sum of the reciprocal eighth powers, tenth powers, twelfth powers, with the observation that this took a fair deal of work for the higher powers.