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$$S_n=\sum_{k=1}^{n} \frac{k}{k^2+n^2}$$ $$U_n=\sum_{k=0}^{n-1} \frac{k}{k^2+n^2}$$ I plotted these curves using desmos and found that $S_n$ was decreasing and $U_n$ was increasing. But I was unable to prove this fact.

Since the limits of the summation and each individual terms include $n$, I cannot directly say that $S_n$ increases (or decreases) as n increases. Similarly for $U_n$.

If I could show that $S_n$ is decreasing, I can convert $S_n$ to the limit of a sum as n tends to infinity and convert it to a definite integral, thus finding the lower bound for $S_n$, and using a similar argument for $U_n$, the upper bound of $U_n$.

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  • $\begingroup$ Wild guess: what is S(n+1)-S(n) and U(n+1)-U(n)? $\endgroup$
    – user2469
    Commented Jan 24, 2016 at 2:47

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