where is replaced by after the right-hand side has been expanded symbolically.
They can also be derived as coefficients in the following power series expansion:
The hopeful fact about Bernoulli polynomials in our case is that
which means that the equation whose solutions we are investigating,
boils down to
Now is a polynomial of degree n+1 in x, so that in the area we’re looking in, where x-n is approximately 0.3x, will be infinitesimal in comparison with , so that the simplified equation
has the same asymptotic behaviour as the original.
Another hopeful line of inquiry is that is the coefficient of in the power series expansion of , while is the coefficient of in the expansion of .
Unfortunately the the x we are looking for is not constant but depends on n, so that we can’t establish interesting facts about and as a whole. All we can really do to isolate a particular power of t is to differentiate each of these two functions n times with respect to t and compare the results at t=0. Nevertheless, it is still good to find some sort of an occurrence of an exponential function, given that we are trying to get a reason for ln 2 appearing in the result.