Are you trying to derive the formula for , the nth Bernoulli number?.

For the Bernoulli polynomials, we have:

The resulting polynomials are our Bernoulli polynomials.

And are denoted by

is a poly with degree n with rational coefficients. The first few are:

..............[1]

To display the Bernoulli polys in a form in which their coefficients become nice and elegant, we can write the constant terms as:

Then we get:

and so on.

in general:

............[2]

This ca be verified by induction.

But, as we know, are the Bernoulli numbers.

So, from [1], we see we get:

Their computation can be found by the observation that, for n=1,2,3,...

and therefore

Thus, bu [2]:

This gives us a recursive procedure for finding [tex]B_{n}[tex] once are known.

So, when n is odd and greater than 1. And alternates in sign whenever n is even.

Now, we can derive the formula for the sum of consecutive powers.

For every real number x,

When n=0, the right side is equal to 1.

Proceeding with induction, we assume its tre for n and show that:

Skipping ahead, we can derive Euler's beautiful formula for the sum of the reciprocals of all even powers:

For k=1,

k=2,

k=3,

There is much more, but I am tired of typing. I may come back with more. I done research on these in undergrad some time back. I hope this was a nice tutorial.