First, thanks for your previous help. It was greatly needed and appreciated.

So we have proved [A,B^{n}]=nB^{n-1}[A,B]

Next we have: If F(x)= is any Taylor Series expandable function with an appropriate radius of convergence, and A and B satisfy the conditions already stated, show that the operator function F(B) obeys , where .

Using earlier proof

Now the last part: Use prior result to evaluate where L is a constant having units of length.

I got

Would someone be able to verify the second part and the last part?

Thank you