I need help finding the limit of
e^(-n) * (1 + 1/n)^(n^2)
That's the same "error" that TD pointed out (if error it is). And it appears I lied... The expression with n = 1000 (as far as my calculator will go) appears to be decreasing but fairly flat (first derivative at n = 1000 is -2 x 10^(-7)) and only at 0.6 or so.
I can't say why TPH and Soroban's methods would be wrong, but I don't think the answer is 1.
However, I am confused why my results produced the incorrect answer. It is certainly that the limit composition rule failed, for some reason. Perhaps, when I said it works for was wrong, I know that it works for numbers, I just always assumed it works for infinite limits also. Apparently it seems by assumption was wrong.
Though I can prepared to be wrong, and think I am. What you shown was not a proof . It needs to be formal.
EDIT. I realized my mistake.
What I said about composition functions holds true. But I have not properly expressed the outer function, I assumed it was raised to the , but that does not succesfully express the outer function.
It looks correct to me. Again as I said the limit composition rule is used here.
The only think I am concerned about is whether a distribution of the limit to each individual term is allowed in an infinite series. I think yes, I believe you can always do it. Or whether the series need to be absolutely convergent, but I think you are correct about this one.
It some ways I agree with that approach. I think I am the only mathemation in my differencial equations class. I have made some remarks about the professors' solution after class, he told me he would be concered with it if he was in a roomful of mathemations but engineers do not need to know that stuff. In fact, If I want to for example, differencial equations on a serious level I would first learn how to solve them informally and incorrectly. Then I would learn the theory, so that I not have to memorize techniques to how to solve them, that stuff I would already know and only be concerned with about the theory.