Given functions $f:\Bbb Z_+\to \Bbb Z_n $ and $g:Z_+\to \Bbb Z_m$ and suppose
$$\displaystyle\sum_{k=1}^\infty f(k)\cdot n^{-k}=\sum_{k=1}^\infty g(k)\cdot m^{-k}$$
Is there a method to express $f(k)$ with $\{g(i)\}_{i\in\Bbb Z_+}$ for each k?
That is, how to convert infinite fractions with different number basis?
I now how to do to convert finite sequences, e.g. $0.321_8$ to decimal $(12_8=10_{10})$:
321x12= 4052 4
052x12= 0644 0
644x12=10150 8
150x12= 2020 2
020x12= 0240 0
240x12= 3100 3
100x12= 1200 1
200x12= 2400 2
400x12= 5000 5
000x12= 0
So $0.321_8=0.408203125_{10}$