We know that
For higher powers, the expansion gets very tedious by hand! Fortunately, the Binomial Theorem gives us the expansion for any positive integer power of (x+y): Binomial Theorem For any positive integer n,
Example By the Binomial Theorem,
Extensions of the Binomial Theorem A useful special case of the Binomial Theorem is
This formula can be extended to all real powers α:
Notice that the formula now gives an infinite series. (When α = n is a positive integer, all but the first (n+1) terms are 0 since after this n-n ( = 0) appears in each numerator.)
This expansion is very useful for approximating
(1+x)α for
|x|| << 1:
But for |x| << 1, higher powers of x get small very quickly, so (1+x)α can be approximated to any accuracy we need by truncating the series after a finite number of terms. Example For |x| << 1,
This type of reasoning is useful in investigating what happens when a physical system is perturbed slightly, introducing a new very small term x. Binomial Theorem For any positive integer n,
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