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Given $a_n = \sum\limits_{k=1}^{n} \left(\sqrt{1+\frac{k}{n^2}}-1\right)$, find $\lim\limits_{n \to \infty} a_n$.

My try: To simplify, $$a_n = \frac{\displaystyle\sum_{k=1}^{n}\sqrt{n^2+k}-n}{n}$$ and I'm stuck from there. In addition, I have made a program to find the limit, which says it's 1/4. Can anybody give me a hint to start? Thanks for your time!

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    Related (if not a duplicate): [Prove that limit $\lim_{n\to\infty}\sqrt{4+\frac{1}{n^2}}+\sqrt{4+\frac{2}{n^2}}+\cdots+ \sqrt{4+\frac{n}{n^2}}-2n=\frac{1}{8}$](http://math.stackexchange.com/questions/833824) – 2017-01-14
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    Hint: Use the identity $$\sqrt{1+\frac{k}{n^2}}-1=\frac{\frac{k}{n^2}}{\sqrt{1+\frac{k}{n^2}}+1}$$ which implies that, for every $1\leqslant k\leqslant n$, $$\frac{1}{1+\sqrt{1+\frac1n}}\cdot\frac{k}{n^2}\leqslant\sqrt{1+\frac{k}{n^2}}-1\leqslant\frac12\cdot\frac{k}{n^2}$$ – 2017-01-14
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    the searched limit is $$\frac{1}{4}$$ – 2017-01-14

1 Answers 1

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Generalization

Suppose $f$ is differentiable at $x=0$ with $f(0)=0$ and consider, for all $n\in\mathbb{N}^\star$ :

$$S_n=\sum_{k=1}^nf\left(\frac{k}{n^2}\right)$$

Then we have :

$$\lim_{n\to\infty}S_n=\frac{1}{2}f'(0)$$

It's relatively easy to prove this, using a Taylor expansion. We know that :

$$f(x)=\underbrace{f(0)}+xf'(0)+x\alpha(x)$$

with $\lim_{x\to 0}\alpha(x)=0$. Hence :

$$S_n=\frac{f'(0)}{n^2}\sum_{k=1}^{n}k+\frac{1}{n^2}\sum_{k=1}^nk\,\alpha\left(\frac{k}{n^2}\right)$$

The first piece has limit $\frac{1}{2}f'(0)$ because $\sum_{k=1}^nk=\frac{n(n+1)}{2}$.

We prove now that the second piece has limit $0$ :

Given $\epsilon>0$, there exists $\delta>0$ such that :

$$\forall x\in\mathbb{R},\vert x\vert\le\delta\implies\left|\alpha(x)\right|\le\epsilon$$

If $n$ is large enough (and $n\ge\frac{1}{\delta}$ is enough), we have :

$$\forall k\in\{1,\cdots,n\},\,0\le\frac{k}{n^2}\le\frac{1}{n}\le\delta\;\mathrm{and}\;\mathrm{therefore}\;\left|\alpha\left(\frac{k}{n^2}\right)\right|\le\epsilon$$

so that :

$$\left|\frac{1}{n^2}\sum_{k=1}^nk\,\alpha\left(\frac{k}{n^2}\right)\right|\le\frac{1}{n^2}\sum_{k=1}^nk\,\left|\alpha\left(\frac{k}{n^2}\right)\right|\le\frac{\epsilon}{n^2}\frac{n(n+1)}{2}=\frac{\epsilon(n+1)}{2n}\le\epsilon$$