Limits vs Summation

Calculus Level 3

limn(n2b+12n2a1)= 1 \large \displaystyle\lim_{n\to\infty}\left(\dfrac{n^2b+1}{2n^2a-1}\right)=\ 1

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

What is the value of n=1(aba2+b2)n ?\displaystyle\sum_{n=1}^{\infty}\left(\dfrac{ab}{a^2+b^2}\right)^n \ ?

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