Help with power series solution

[mmwave]

I've done a power series solution to a differential equation and got the recursive formula for the coefficients below. Now I am to evaluate it for large j and I don't get the answer in the book.

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 )

 

[HallsofIvy]

Essentially, they are ignoring terms "small" compared with the very large j: in this case that's just about everything that doesn't involve j.

I must admit that I don't understand why they include the "+1" in "j+1" in the denominator. If j is large, then 1 is certainly small compared with it. Also, there is no reason not to cancel the js in the numerator and denominator.

If j= 10000, say, then 2j/(j(j+1))= 20000/(10000*10001)= 20000/100100= 0.00019998 while 2/(j+1)= 2/(10001)= 0.00019998- exactly the same thing and 2/j= 2/10000= 0.0002.

Your use of L'Hopital's rule IS incorrect. Your first result of
aj * 2 (j) / ( j * j) = aj * 2/j is good.

 

[M98Ranger]

It seems like you have done a great job with your power series solution and have come up with a valid recursive formula for the coefficients. In order to evaluate it for large j, you have correctly considered the limit as j approaches infinity. However, the book's answer may be using a different approach or simplifying the expression in a different way.

To get to their answer, they may have used the fact that (j+L+1) can be approximated as j when j is large. This is because the constant L would have a much smaller contribution compared to the rapidly increasing j. Similarly, they may have approximated (j+2L+2) as j+1. This would result in their expression of aj+1 approx. aj * 2j / (j*(j+1)), which is equivalent to aj * 2/j as j approaches infinity.

It's also possible that the book is using a different method or technique to solve the differential equation, which is why their answer may differ from yours. It's important to remember that there can be multiple valid ways to solve a problem, and the key is to understand the reasoning behind each step.

In summary, both your approach and the book's approach are valid and may just differ in their simplifications or approximations. Keep up the good work and continue to ask questions when you encounter discrepancies in your solutions.