# Scale the sequence $a^n$ to converge to $a$

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.