I've been playing around with infinite sums for quite a while now, but recently, I've come across the following question in an Under-Graduate Mathematics Book that is specifically targeted at problem solving. The problem is as follows,
Evaluate the following:
$$\sum_{r=2}^\infty \Biggl(\binom{2r}{r}{\biggl(\frac{x}{4}\biggr)^r}\Biggr)^2$$
where x is strictly less than unity.
I've thoroughly checked that the sum is, in fact, convergent, however, I am completely stumped as to how I am to evaluate it. I am guessing that the final expression is one involving the variable 'x' since I do not see any way for it to be eliminated somehow. Any kind of hint/solution/explanation to the problem would be highly appreciated.
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