Hello, Espionage!
The geometric series has first term and common ratio
Its sum is: .
Hence: .
Therefore: .
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If I was going to find the infinite sum, I would begin with the difference equation to obtain the partial sum in closed form:
where
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:
And thus:
By superposition, we have:
Using the initial value, we find:
and so we find:
Hence, the infinite sum is: