Word Count : 238

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We are asked to evaluate $$ \lim_{n\to\infty} \left[ \frac{n+1}{n^2+1^2} + \frac{n+2}{n^2+2^2} + \frac{n+3}{n^2+3^2} + \cdots + \frac{n+n}{n^2+n^2} \right] $$ using Cauchy’s integral test (which essentially compares a sum with an integral in the limit). Let’s solve step by step. --- Step 1: Rewrite the sum in a general form Let the sum be $$ S_n = \sum_{k=1}^{n} \frac{n+k}{n^2+k^2}. $$ ____ ______ _____ __________ ___ ____ _________.
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