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426 Rh CHAPTER XLV.

Binomial Theorem. Any Index.

531. In Chapter xxxvii. we investigated the Binomial Theorem when the index was any positive integer; we shall now consider whether the formul there obtained hold in the case of negative and fractional values of the index.

Since, by Art. 411, every binomial may be reduced to one common type, it will be sufficient to confine our attention to binomials of the form $$(1+x)^n$$.

By actual evolution we have

$$(1 + x)^{\frac{1}{2}}$$ =

and by actual division,

$$(1 + x)^{-2}$$ =

and in each of these series the number of terms is unlimited.

In these cases we have by independent processes obtained an expansion for each of the expressions $$(1 + x)^{\frac{1}{2}}$$ and $$(1 + x)^{-2}$$ We shall presently prove that they are only particular cases of the general formula for the expansion of $$(1 +x)^n$$, where n is any rational quantity.

This formula was discovered by Newton.

532. Suppose we have two expressions arranged in ascending powers of x, such as

and