05/01/2015, 09:43 PM
Assuming a_n Goes to < (2/3)^n ;
( this gives us a sufficiently Large radius such that the equation is satisfied within the ROC.)
Taylors theorem gives us
f(x+1) = f(x) + f ' (x) + f " (x)/2 + ...
Hence what the truncation of degree k solves locally is near ;
f(x+1) + O(a_k x^k) = exp(f(x))
f(0)=1
By taking k Large and x small we get :
f(0)=1
f(x+1)=exp(f(x)) + o(f(1)).
( take x < 1 to see this )
Notice lim o(f(1)) = lim a_k = 0.
Hence we have in the limit k to oo assuming the ROC ;
f(0)=1
f(x+1) = exp(f(x))
Qed
So the attention Goes completely to the asymp of a_n.
Hope that is clear.
Q: Can we show existance and uniqueness for these equations FORMALLY ?
Q2 ; i Will post in a new thread.
Regards
Tommy1729
( this gives us a sufficiently Large radius such that the equation is satisfied within the ROC.)
Taylors theorem gives us
f(x+1) = f(x) + f ' (x) + f " (x)/2 + ...
Hence what the truncation of degree k solves locally is near ;
f(x+1) + O(a_k x^k) = exp(f(x))
f(0)=1
By taking k Large and x small we get :
f(0)=1
f(x+1)=exp(f(x)) + o(f(1)).
( take x < 1 to see this )
Notice lim o(f(1)) = lim a_k = 0.
Hence we have in the limit k to oo assuming the ROC ;
f(0)=1
f(x+1) = exp(f(x))
Qed
So the attention Goes completely to the asymp of a_n.
Hope that is clear.
Q: Can we show existance and uniqueness for these equations FORMALLY ?
Q2 ; i Will post in a new thread.
Regards
Tommy1729
