I would like to understand the asymptotic behaviour of a linear combination of functions and their derivatives, where at least one diverges. This problem appears naturally if one wants to compute the evolution of specific couplings in Non-Abelian Gauge Theory.
Given $F_1 ,\dots F_N $ analytical functions on $[\ 0,R [\ $, and the real sequence $a=\left(a_{i,j}\right)_{0\leq i \leq j}$ we defined:
$\forall n \geq N $ (integer) and $ \forall x \in [\ 0, R [\ $:
$\mathcal{V}_{n,a} \left(x \right) = \sum_{i=1}^{N} x^{n-i} \left[ \sum_{j=0}^{n-i} a_{j,n-i} x^j {F_i}^{ \left( j \right)} \left( x\right) \right] $
I only have a partial knowledge on $F_1$ and the $ a_{i,j}$'s :
the asymptotic condition ${F_1}\left(x\right) \sim_{x \rightarrow R} c_0 \; \ln\left(R -x \right) $ with $c_0>0$
And $\forall k \geq 1, \exists c_k>0 $, ${F_1}^{\left(k\right)}\left(x\right) \sim_{x \rightarrow R} c_k\frac{\left(-1\right)^{k+1} }{\left(x-R\right)^k} $
$a_{i,i}=\frac{t^{i-1}}{\left(i-1\right)!}$ with $ t \in {\mathbb{R}}^*$
As we can see, $ \mathcal{V}_{n,a} $ contains ${F_1}^{\left(n-1 \right)}$ whose divergence is greater with $n$, so here is the question:
-1)Does it exist $n_{min}\geq N $ and $a$ (and so $t$), such as $ \forall n \geq n_{min}, \mathcal{V}_{n,a}\left(x\right)$ be bounded in a neighbourhood of $R$.
-2)If no, it means that there is a strictly increasing sequence of integer $n_i, i \in \mathbb{N}$ for which $\mathcal{V}_{n_i,a}$ will not be bounded . Then in a neighbourhood of $R$, should we expect $\mathcal{V}_{n_i,a}$ to dominate over $\mathcal{V}_{n_j,a}, for j<i$?
What I tried: a specific case with $N=2$ and $F_2$ a Laurent series of the form $F_2\left(x\right)= \sum_{k\geq -1} b_k\left(x-R\right)^k $. It appears that $\mathcal{V}_{n,a}$ can never be bounded and that that indeed $\mathcal{V}_{n_i,a}$ dominates over $\mathcal{V}_{n_j,a}, j<i$. However that's too specific and I would like to solve 1) and 2) in the general cases.