I'm hoping to solve the system of equations $$ a_1 + a_2 + a_3 + a_4 + ... = A_1$$ $$ a_1 + 2a_2 + 4a_3 + 8a_4 + ... = A_2$$ $$ a_1 + 3a_2 + 9a_3 + 27a_4 + ... = A_3$$ $$ ...$$ $$ a_1 + n^1a_2 + n^2a_3 + n^3a_4 + ... A_n$$ Here, all $A_n$ are known, and there are countably infinitely many $A_n$ given. My hope is to be able to compute alternating versions of the original sums in terms of $A_1$ through $A_n$. So, I'd like to compute $$ a_1 - a_2 + a_3 - a_4 + ... = ??$$ $$ a_1 - 2a_2 + 4a_3 - 8a_4 + ... = ??$$ etc.
I've tried looking at this as a matrix, but I'm unsure of how to invert an infinitely large matrix. Any help would be appreciated!
Edit: As the comments mentioned, the question posed doesn't have a unique solution. Instead, I'd like to look at assigning a unique value by looking at the limit of solutions of n x n matrices. For instance, I can notice a few patterns when looking at the solutions of smaller. For instance for the first two, the top and bottom lines seem to have a pattern. The top lines are values of Pascal's triangle with some sign changes. The bottom lines appear to be 1/factorials with sign changes.
These patterns seem to keep for larger matricies
Is there a nice solution for an arbitrarily large nxn matrix?