In the previous section, we discussed the expansion of , where is a natural number. We’ll extend that discussion to a more general scenario now. In particular, we’ll consider the expansion of , where is a rational number and . Note that any binomial of the form can be reduced to this form.:
(we are assuming )
The general binomial theorem states that
That is, there are an infinite number of terms in the expansion with the general term given by
For an approximate proof of this expansion, we proceed as follows: assuming that the expansion contains an infinite number of terms, we have:
Putting gives . Now differentiating once gives
Putting gives .
Proceeding in this way, we find that the coefficient is given by
Note that if is a natural number, then this expansion reduces to the expansion obtained earlier, because becomes , and the expansion terminates for . For the general , we obviously cannot use since that is defined only for natural .
One very important point that we are emphasizing again is that the general expansion holds only for .
Let us denote the genral binomial coefficient by . Thus, we have
Let us discuss some particularly interesting expansions. In all cases,