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The binomial theorem states that

$(A+B)^n=\sum_{k=1}^{n}{n \choose k}A^{n-k}B^k$

I need help expressing the following summation as a modified binomial expression:

$\sum_{k=1}^{n}n!{n-1\choose k}A^{k+1}B^{n+k}=(A?B)^?$

Thanks

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    You need "$k=0$" rather than "$k=1$".2012-01-04

2 Answers 2

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Your first expression should be $(A+B)^n=\sum_{k=0}^{n}{n \choose k}A^{n-k}B^k$ starting from $k=0$.

Your second expression may have problems with $k=n$ when evaluating ${n-1 \choose n}$.

So I will try to help with a slightly altered version of your question: $\sum_{k=0}^{n-1}n!{n-1\choose k}A^{k+1}B^{n+k} = n!A B^n \sum_{k=0}^{n-1}{n-1\choose k}1^{n-1-k}(AB)^{k} = n!A B^n (1+AB)^{n-1}.$

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    It is there to make it obvious we are applying the earlier binomial expansion of $(1+AB)^{n-1}$ . It can be ignored if necessary2012-01-04
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$ \sum_{k=1}^n n! \binom{n-1}{k} A^{k+1} B^{n+k} = n! \cdot A \cdot B^n \cdot \sum_{k=1}^{n-1} \binom{n-1}{k} A^{k} B^{k} = n! \cdot A \cdot B^{n} \left( (1+A B)^{n-1} - 1 \right) $