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Possible Duplicate:
Evaluating $ \lim_{n\to\infty} \sum_{k=1}^{n^2} \frac{n}{n^2+k^2} $

I recently encountered this question which stumped me because it wasn't obvious what sort series manipulations I could do to find its value. The question is what is the limit as $n$ tends to infinity of $$\sum_{k=0}^{n} \frac{k}{n^2+k^2}$$

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    The body and title ask two different questions. The question in the body makes sense, whereas the title asks about the exact value of a divergent series.2012-11-07
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    See the respective answer. Just change $N=n^2$ to $N=n$ and $f(x)=n/(n^2+x^2)$ to $f(x)=x/(n^2+x^2)$. Then make some computations.2012-11-07
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    @Norbert: At least one needs to find the antiderivative of this - different- $f$, so I won't call it a real duplicate.2012-11-07

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$$\lim_{n \to \infty} \sum_{k=0}^n \dfrac{k}{n^2 + k^2} = \lim_{n \to \infty} \sum_{k=0}^n \dfrac{1}{n} \dfrac{\dfrac{k}{n}}{1 + \left(\dfrac{k}{n}\right)^2}.$$

The right side is a Riemann sum: The interval $[0, 1]$ is divided into $n$ subintervals of width $\dfrac{1}{n}$, and rectangles of height $\dfrac{\dfrac{k}{n}}{1 + \left(\dfrac{k}{n}\right)^2}$ are constructed on the subintervals. The sum is the sum of the areas of the rectangles, an approximation to the area under $f(x) = \dfrac{x}{1 + x^2}$ from $x = 0$ to $x = 1$. The limit lets the number of rectangles go to infinity. This gives the integral

$$\int_0^1 \dfrac{x}{1 + x^2}\,dx = \dfrac{1}{2} \ln 2.$$