Prove that:
$$b_n = 1 + \sum\limits_{k=1}^{∞} \binom{n-1}{k}b_k.$$
Workings:
The first thing I noticed is that the above equation looks very similar to a Bell Numbers proof:
$b_{n+1}=\sum\limits_{k=0}^n\binom{n}{k}b_k$
Making me think there is some sort of relation between the two. Though this may not be true. Since the one I need to prove does have an infinity.
So because of this I'm not to sure on what to do.
Any help will be appreciated.
If $k > n-1$ then $\binom{n-1}{k} = 0$. Thus the original equality states $$b_n = 1 + \sum_{k=1}^{n-1} \binom{n-1}{k} b_k.$$ If you have that $b_0 = 1$ then $1 = \binom{n-1}{0} b_0$ so in fact $$b_n = \sum_{k=0}^{n-1} \binom{n-1}{k} b_k.$$