Binomial theorem
The binomial theorem with a negative exponent is the fraction
where n is a natural number.
Explanation
We write the binomial expansion in detail and see
The development of the binomial coefficients for negative exponents is according Pascal's triangle. A number is the sum of the number on the left and the right, just above it. You see it at −4 = −5 + 1 in the row before last.
n 0 1 2 3 4 5 6 7 8 k → ∞ 8 1 8 28 56 70 56 28 8 1 7 1 7 21 35 35 21 7 1 ◦ 6 1 6 15 20 15 6 1 ◦ ◦ 5 1 5 10 10 5 1 ◦ ◦ ◦ 4 1 4 6 4 1 ◦ ◦ ◦ ◦ 3 1 3 3 1 ◦ ◦ ◦ ◦ ◦ 2 1 2 1 ◦ ◦ ◦ ◦ ◦ ◦ 1 1 1 ◦ ◦ ◦ ◦ ◦ ◦ ◦ 0 1 ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ −1 1 −1 1 −1 1 −1 1 −1 1 ··· −2 1 −2 3 −4 5 −6 7 −8 9 ··· −3 1 −3 6 −10 15 −21 28 −36 45 ··· −4 1 −4 10 −20 35 −56 84 −120 165 ··· −5 1 −5 15 −35 70 −126 210 −330 495 ···
It is evident that for negative exponents every row has an infinit number of terms. There emerge alternating positive and negative coefficients, that increase in value. For negative exponents the formule is
The upper number n is the exponent of the binomial theorem, the bottom number k is the current number of the term in the result. You can not use this formula for positive exponents. We can write
You can see for yourself that
Example 1
For n = 4 the formula gives
We can only use these series for a = 1, because then
If b ≤ 1 a converging series arises