This series is supposed to be an infinite series but it's not going up by a set common ratio. How do I find the sum of the infinite series of this series?
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: