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Find $$ \lim\limits_{n\to\infty}\sum\limits_{i=1}^{n}{\frac{2n}{(n+2i)^2}}.$$

I have tried dividing through by $1/n^2$ and various other algebraic tricks but cannot seem to make any progress on this limit. Wolfram Alpha gives the value as $2/3$, but I could not understand its derivation. Any insight is welcome.

2 Answers 2

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The expression $$ \sum_{i = 1}^n \frac{2n}{(n+2i)^2} =\frac{1}{n} \sum_{i = 1}^n \frac{2}{(1+\frac{2i}{n})^2} $$ is the Riemann sum of the function $f(x)= \frac{2}{(1+2x)^2}$ over the interval $I = [0,1]$, corresponding to the uniform partition of $I$ into $n$ equal parts. Since $f$ is Riemann-integrable (being a continuous function over a closed and bounded interval), this sum approaches the integral of $f$ over $I$ as $n \to \infty$. That is, $$ \begin{eqnarray*} \lim_{n \to \infty} \frac{1}{n} \sum_{i = 1}^n \frac{2}{(1+\frac{2i}{n})^2} &=& \int_0^1 \frac{2}{(1+2x)^2} \mathrm{d} \, x \\ &=& \left. -\frac{1}{1+2x} \right|_{x=0}^{x=1} \quad = \quad \frac{2}{3}. \end{eqnarray*} $$

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    Thanks @Daniel, for coming back to accept an answer. Many users forget to do that... =)2011-10-31
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You can also use Euler-Maclaurin summation.

The first-order Euler-Maclaurin formula says $$\sum_{i=1}^n f(i) = \int_1^n f(x) \, dx + {f(1) + f(n) \over 2} + \int_1^n f'(x) \left(x - \lfloor x \rfloor - \frac{1}{2}\right)\,dx.$$

Since $|x - \lfloor x \rfloor - \frac{1}{2}| \leq 1$, with $f(x) = \frac{2n}{(n+2x)^2}$ we have $$\sum_{i=1}^n \frac{2n}{(n+2i)^2} = \int_1^n \frac{2n \, dx}{(n+2x)^2} + R_n,$$ where $|R_n| \leq \left|\frac{3f(n) - f(1)}{2}\right| = \left|\frac{3}{n} - \frac{n}{(n+2)^2}\right|$.

Therefore, $$\lim_{n \to \infty} \sum_{i=1}^n \frac{2n}{(n+2i)^2} = \lim_{n \to \infty} \int_1^n \frac{2n \, dx}{(n+2x)^2} = \lim_{n \to \infty} \left[\frac{-n }{n+2x} \right]_1^n = \lim_{n \to \infty} \left(-\frac{1}{3} + \frac{n}{n+2}\right) = \frac{2}{3}.$$

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    +1 Your solution is also nice. I especially like the fact that this yields an error bound of $O(1/n)$ as well.2011-10-29
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    This solution is good, but we need the best one which is simpler than this one.2011-10-29
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    @Hassan: I am missing the point of your comment...2011-10-29