I was recently wondering if there was some method to "flip" the series and still assign it a value for divergent series. To illustrate what I mean:
$$ S(x,n) = a_0 + a_1 x + a_2 x^2 + \dots + a_n x^n $$
We would like to define the flipped sum series:
$$ F(S(x,n)) = a_n + a_{n-1} x^{1} + \dots + a_0 x^{n} $$
Hence, to do this we begin by: assigning $x \to x^{-1} $ and take $x^n$ common:
$$ S(x^{-1},n) = (a_n + a_{n-1} x^{1} + \dots + a_0 x^{n}) x^{-n} $$
Multiplying both sides with $x^n$:
$$ S(x^{-1},n) x^n = (a_n + a_{n-1} x^{1} + \dots + a_0 x^{n})$$
Taking limit $n \to \infty$:
$$ \lim_{n \to \infty}S(x^{-1},n) x^n = \lim_{n \to \infty}(a_n + a_{n-1} x^{1} + \dots + a_0 x^{n}) $$
Taking the second limit $x \to 1$:
$$\lim_{x \to 1 } \lim_{n \to \infty}S(x^{-1},n) x^n = \lim_{n \to \infty} (a_n + a_{n-1} + \dots + a_0 )$$
Questions
Is the above proof correct? Can this somehow be extended to zeta regularization?