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Let \(k\) be such that \(f(k)=8\), Then \(g(k)=3\) and we get \[3=k(x-a)(x-b)(x-c)(x-d)h(x).\] By the fundamental theorem of arithmetic, we can express 3 as a product of at most three different integers \((-1,-3,1)\).
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How would you evaluate the following series? $$\lim_{n\to\infty} \sum_{k=1}^{n^2} \frac{n}{n^2+k^2} $$ Thanks.