Binomial theorem
If the first variable of the binomial theorem is 1, the power series development becomes
for each whole, rational and negative exponent.
Explanation
From the fundamentel theorem of mathematics follows that you can write the function f (x) = (1 + x)n as
and for x = 0 it gives 1 = a0. The first derivative is
and for x = 0 it gives n = a1. The second derivative is
and for x = 0 it gives n(n − 1) = 2a2. By substitution of a0, a1 and a2 you get
Example 1
For negative exponents, geometric progressions are created
(1 + x)−1 = 1 − x + x2 − x3 + x4 − x5 + x6 − x7 + ···
(1 + x)−2 = 1 − 2x + 3x2 − 4x3 + 5x4 − 6x5 + 7x6 − 8x7 + ···