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$\large \displaystyle\lim_{n\to\infty}\left(\dfrac{n^2b+1}{2n^2a-1}\right)=\ 1$

Suppose for real numbers $a,b$, we have the limit above.

What is the value of $\displaystyle\sum_{n=1}^{\infty}\left(\dfrac{ab}{a^2+b^2}\right)^n \ ?$

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