Can i find or assume the existence of a sequence $f_n$ that converges the divergent sequence $a^n$ or the sequence $e^{a\cdot n} $ when they are multiplied?

I can write this problem as:

Find $f_n$ such that the sequence

$g_n = f_n\cdot a^n$

converges to $a$.

Or similarly find $f_n$ such that

$h_n = f_n \cdot e^{a\cdot n}$ converges to $a$.

Choices like $f_n = 1/a^{n-1}$ or $f_n = \exp(-a)$ are not acceptable.

The main idea is to find a sequence of kernels $k_n(x,y) = f_n \sum_i (\mathbf{x}^T P_i \mathbf{y})^n$ or $k_n(\mathbf{x},\mathbf{y}) = f_n \sum_i \exp(n\mathbf{x}^T P_i \mathbf{y})$ that converge to the maximum value $\lim_{n\rightarrow \inf}(k_n(\mathbf{x},\mathbf{y})) =\max_i(\mathbf{x}^T P_i \mathbf{y})$.

Without $f_n$ these sequences converge to $(\max_i(\mathbf{x}^T P_i \mathbf{y}))^n$ and $\exp(n\max_i(\mathbf{x}^T P_i\mathbf{y}))$ respectively,

where $P_i$ are all the permutations of the vector y.



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