Hey guys:

I used three ways to calculate a list of value based on a recursive formula. For the last method, I calculated the theoretical result first and then feed in the numerical value. However, it turns out numerical result is significantly different for three ways, which really surprises me. I think, at least, first two should be the same as the only difference is that I do not create a list to save and quote for the first method.

Really appreciate any comments for the cause of this and how to improve.

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