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