For any positive integer , find

“The following elegant solution [says Dörrie] is based upon the binomial theorem.

By resorting to the device of considering the magnitudes resulting from the binomial expansion of as unknowns subject to certain conditions rather than as powers of , we obtain an amazingly short derivation of .”

(This type of crazy-looking procedure is Blissard’s symbolic method, or umbral calculus, introduced by John Blissard in 1861. Also used extensively by Lucas. Now referred to as 19th C or classic umbral calculus, as since the 1970s the subject has been transformed by Roman, Rota and others.)

According to the binomial theorem, if ,

and

Since subtracting the second equation from the first gives:

Now define the unknowns by the equations

etc.

This simplifies to

Substitute in to get

Addition of these equations gives

or

Now to determine from equations , ,

From it follows that

From ,

From ,

From ,

These are known as the Bernoulli numbers:

Then from we get