multinomial theorem (proof)
First, for , both sides equal . For the induction step, suppose the multinomial theorem holds for . Then the binomial theorem and the induction assumption yield
where and is a multi-index in . To complete the proof, we need to show that the sets
are equal. The inclusion is clear since
For , suppose , and . Let . Then , so for some . It follows that that .
Let us define and let be a multi-index in . Then
This completes the proof.
|Title||multinomial theorem (proof)|
|Date of creation||2013-03-22 13:41:55|
|Last modified on||2013-03-22 13:41:55|
|Last modified by||Koro (127)|