Consideration of the binomial theorem for fractional index, or of the continued fraction representing a surd, or of theorems such as Wallis's theorem (ï¿½ 64), shows that a sequence, every term of which is rational, may have as its limit an irrational number, i.e.
(iv.) The procedure is sometimes stated differently, the transposition being regarded as a corollary from a general theorem that the roots of an equation are not altered if the same expression is added to or subtracted from both members of the equation.
ï¿½ 21 (ii.)) is that we do not need the general theorem, and that it is unwise to cultivate the habit of laying down a general law as a justification for an isolated action.
The binomial theorem gives a formula for writing down the coefficient of any stated term in the expansion of any stated power of a given binomial.
(ii.) We can prove the theorem of ï¿½ 41 (v.) by a double application of the method.