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I'm not sure what method they are using to get the answer although their answer makes sense physically.

a j+1 = aj * 2 * {(j + L + 1) - k }/ ( {j+1}(j + 2L + 2) )

where L and k are constants and j is just an integer index number.

If I consider large j I would say

aj+1 approx. aj * 2 (j) / ( j * j) = aj * 2/j

If I say j => infinity and use l'Hopital's rule I get

aj+1 = aj * 2 / (2j + 2L + 3) approx. aj * 1/j

The book gets

aj+1 approx. aj * 2j / ( j*(j+1))

What rules are they applying to get this?

(of course a j+1 means a

_{j+1}and aj means a

_{j})