The Cauchy Product Formula tells us that
From last time we got
So using the formula from above we get
Since j does not depend on k, we can multiply the rightmost sum by and divide by that in the inner sum.But when doing that, we get a binomial in the right sum
When j=0, the output is 1(because must be 1), so the sum of all the others is exactly 0.But we defined z not to be equal to 0, so
Thus we can find a relation between the 's and , as we know , we can find all terms in the sequence (these numbers are called Bernoulli numbers).
So now we have found the terms in the infinite series for .
Note that the function is even, thus all the powers in its series must be even, so for .
Now let's rewrite that sum of ours, starting from n=2.
But , so the first and the third cancel.Also, , so we can again rewrite our sum
Now let's put a in the sum so we get a -2 up front.
From Part 2 we know that
So comparing the coefficients we get that