Finding roots of a non linear expression when multiplied with a linear expression
I don't think this equation can be solved in general. Since it is an infinitely long polynomial, it cannot be evaluated. It can only be done if the constants G_t:t+n followed a very specific pattern, so that the series becomes equivalent to a known function that we can evaluate. For example:


If $G_{t:t+n}$ were equal to [ 1, 1/2!, 1/3!, 1/4!, ...] (where 4! is the factorial of 4), then the sum would be equal to $e^\lambda$
If $G_{t:t+n}$ were equal to [1, 0, -1/3!, 0, 1/5!, ...] then the sum would be equal to $\sin(\lambda)$


But in general, if G were any random set of constants, the function cannot be evaluated.