Binomial theorem
For and , .
Theorem 3.13 (Binomial Theorem)
For all and , we have
Proof
When we expand , we obtain terms of the form where . The number of terms of the form in the expansion is , since we must specify brackets from which to pick . Hence
A useful expansion
Setting and gives
Thus, for a small , a good approximation to is — the first-order approximation.
Related
Stated in
- Theorem 3.13 (Binomial Theorem)§3.3.1 Binomial Coefficients