In order to exactly prove Eq. (2.236), one should start by realizing that
arises when one sets n = 0; definition of power of nil exponent and summation, as well as
that entails a universal condition. Suppose now that Eq. (2.236) is valid for a given n; its left‐hand side would then read
(2.251)
for n + 1, where power splitting and application of the distributive property meanwhile took place; insertion of Eq. (2.236) leads to
since it applies to (x + y)n by hypothesis. After factoring x and y, Eq. (2.252) becomes
(2.253)
where the last term of the first summation and the first term of the second summation may to advantage be made explicit as
Equation (2.254) may take the simpler form
because y0 = x0 = 1 and
(2.256)
in view of the similarity of lower and upper limits for the two summations, one may lump them to get
(2.257)
– where xk yn+1−k may, in turn, be factored out as
Equation (2.248) may now be invoked to reformulate Eq. (2.258) to
(2.259)
while the first and last terms may be rewritten to get
(2.260)
association of such terms to the outstanding summation is then fully justified, viz.
If Eq.