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Is my solution right?

\begin{align} \lim_{n\to\infty}\left(\frac{1}{n^{2}}+\frac{2}{n^{2}}+\cdots+\frac{n-1}{n^{2}}\right) &=\lim_{n\to\infty}\left(\frac{1}{n^{2}}\left(1+2+3+\cdots+n-1\right)\right)\\ &=\lim_{n\to\infty}\left(\frac{1}{n^{2}}\left(\frac{(n-1)(1+n-1)}{2}\right)\right)\\ &=\lim_{n\to\infty}\left(\frac{1}{n^{2}}\left(\frac{n(n-1)}{2}\right)\right)\\ &=\lim_{n\to\infty}\left(\frac{n-1}{2n}\right)\\ &=\lim_{n\to\infty}\left(\frac{1}{2}\right)\\ &=\boxed{\frac{1}{2}} \end{align}

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    $\begingroup$ Yeah. absolutely. $\endgroup$
    – MH.Lee
    Oct 20, 2021 at 9:34
  • $\begingroup$ You could say first that you are using the known formula $1+2+\cdots +n-1=n(n-1)/2$. Then the second step can be omitted - which is better. This $1+n-1$ is a bit distracting anyway. $\endgroup$ Oct 20, 2021 at 9:36
  • $\begingroup$ thank you my friend $\endgroup$
    – DanielG
    Oct 20, 2021 at 9:37
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    $\begingroup$ I would skip that $\lim_{n\to \infty} \left(\frac{1}{2}\right)$ bit too. $\endgroup$
    – Gary
    Oct 20, 2021 at 9:41
  • $\begingroup$ @DanielG. You know that you have to '@' the user you are sending the comment in order to notify them right? $\endgroup$ Oct 20, 2021 at 9:54

2 Answers 2

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Correct solution. Another way is to use Riemann sums: \begin{align*} \sum_{k=1}^{n-1}\frac k{n^2} &=\frac 1n\sum_{k=1}^n\frac kn-\frac1n \end{align*} and\begin{align*} \lim_n\left(\frac 1n\sum_{k=1}^n\frac kn\right)=\int_0^1x\;dx=\frac12. \end{align*}

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It is absolutely correct. Although while writing the second line, I think it would be better to write $$1+2+3+\cdots+\color{red}{(}n-1\color{red}{)}$$ instead of $$1+2+3+\cdots+n-1$$ as it may be interpreted as the sum to $n$ and then subtracting a $1$.

Have a good day :)

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    $\begingroup$ I do not see any issue with this answer since the question shows an attempt and is therefore not a low-quality question. Which other reason is there for a downvote ? (+1) $\endgroup$
    – Peter
    Oct 20, 2021 at 11:57

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