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Taylor polynomial. System of equations for the coefficients.
#6
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
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Messages In This Thread
RE: Taylor polinomial. System of equations for the coefficients. - by tommy1729 - 05/01/2015, 09:43 PM

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