Special Cases
If α is a nonnegative integer n, then the (n + 1)th term and all later terms in the series are 0, since each contains a factor (n − n); thus in this case the series is finite and gives the algebraic binomial formula.
The following variant holds for arbitrary complex β, but is especially useful for handling negative integer exponents in (1):
To prove it, substitute x = −z in (1) and apply a binomial coefficient identity.
Read more about this topic: Binomial Series
Famous quotes containing the words special and/or cases:
“When, in some obscure country town, the farmers come together to a special town meeting, to express their opinion on some subject which is vexing to the land, that, I think, is the true Congress, and the most respectable one that is ever assembled in the United States.”
—Henry David Thoreau (1817–1862)
“The world men inhabit ... is rather bleak. It is a world full of doubt and confusion, where vulnerability must be hidden, not shared; where competition, not co-operation, is the order of the day; where men sacrifice the possibility of knowing their own children and sharing in their upbringing, for the sake of a job they may have chosen by chance, which may not suit them and which in many cases dominates their lives to the exclusion of much else.”
—Anna Ford (b. 1943)