This is the true standpoint from which the theorem should be regarded.
This is the binomial theorem for a positive integral index.
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.
These laws can be established either by tracing the individual terms in a sum or a product or by means of the general theorem in ï¿½ 52 (vi.).