I have a countably infinite linear system of equations $Ax = b$, where $A$ is lower triangular with $-1$ at all diagonal entries, and $b = \{-1/2,0,0,...,0\}^T$. I.e the $n$'th unknown depends solely on the entries $A_{i,j}$ where $i\leq n$ and $j\leq n$.
I tried to look at result for the first few variables using a symbolic matrix, but I failed to understand the pattern.
Is it possible to explicitly express the result of the $n$'th unknown using $\prod$, $\sum$, $A_{i,j}$, binomial coefficients etc. without using other unkowns such that it is non-recursive?
You have that $x_1=\frac12$. Then, $x_2=A_{2,1}x_1$, $x_3=A_{3,1}x_1+A_{3,2}x_2=x_1(A_{3,1}+ A_{3,2}A_{2,1})$ and $x_4= \sum_{i=1}^3 A_{4,i}x_i=x_1(A_{4,1}+A_{4,2}A_{3,1}+A_{4,3}A_{3,2}A_{2,1})$ Continuing that way, you obtain: $x_n=\sum_{i=1}^{n-1} A_{n,i}x_i =x_1 \sum_{i=1}^{n-1} \prod_{j=0}^{i-1}A_{n-j,i-j}$