- 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.

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