The geometric series has first term and common ratio
Its sum is: .
If I was going to find the infinite sum, I would begin with the difference equation to obtain the partial sum in closed form:
The corresponding homogeneous solution is:
We then look for a particular solution of the form:
We may now determine the coefficients A and B.
Substituting this into the original difference equation, we find:
Multiply through by :
Equating coefficients, we obtain the system:
By superposition, we have:
Using the initial value, we find:
and so we find:
Hence, the infinite sum is: