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I have an question in compute the following limit: $$ \lim_{n\to \infty} \frac{1}{n} +\frac{2}{n} +...+\frac{n}{n} $$ I know that the answer is: $$ \lim_{n\to \infty} \frac{1}{n} +\frac{2}{n} +...+\frac{n}{n} = \lim_{n\to \infty} \frac{\frac{n(n+1)}{2}}{n} = \infty $$ but a way for compute this limit is: $$ \lim_{n\to \infty} \frac{1}{n} +\frac{2}{n} +...+\frac{n}{n} = \lim_{n\to \infty}\frac{1}{n} + \lim_{n\to \infty}\frac{2}{n} + \ldots + \lim_{n\to \infty}\frac{n}{n} = \lim_{n\to \infty} 0+0+\ldots +1 =1 $$ what second way is not true?

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    $\begingroup$ Try applying your logic to:$$\lim_{n\to\infty}\left(\frac1n+\frac1n+\dotsb+\frac1n\right)$$where there are $n$ terms. On the one hand, you'd think this is equal to $0+0+\dotsb+0=0$. On the other hand, you know that it's equal to $1$ for every value of $n$, so the limit must be $1$! $\endgroup$ Jun 4, 2015 at 1:17
  • $\begingroup$ @amirbahadory I gave you a "+1 vote" for your question and added it to my favorites! $\endgroup$
    – Mark Viola
    Jun 8, 2015 at 3:09

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You can't distribute the limit over a number of summands that depends on $n$, as shown. The first solution is correct. You have $${1\over n }\sum_{k=1}^n k = {n+1\over 2}$$

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In the OP, the conclusion that the limit

$$\lim_{n\to \infty}\left(\frac1 n+\frac2 n+\cdots+\frac{n}{n}\right)=0+0+\cdots +1=1$$

is incorrect.

We can see this more clearly by writing

$$\frac1 n+\frac2 n+\frac3n+\cdots+\frac{n-3}{n}+\frac{n-2}{n}+\frac{n-1}{n}+\frac{n}{n}$$

Now, we see that all of the last terms after the "dots" approach $1$ as $n\to \infty$.

So, in order to approach this problem correctly, we need to write all of the terms as

$$\frac1 n+\frac2 n+\frac3n+\frac4n+\cdots+\frac{n-3}{n}+\frac{n-2}{n}+\frac{n-1}{n}+\frac{n}{n}=\frac1n\sum_{k=1}^n\,k=\frac1n \frac{n(n+1)}{2}=\frac{(n+1)}{2}$$

and now clearly the limit is $\infty$.

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From my point, your method has some question. As is well known, if we add finte terms in the sense of limit, then you can plus every term after you calculate every limit, but if there are infinite terms, you can not do like this.

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You tried something like $$ \lim_{n\to\infty} \sum_{k=1}^n \frac{k}{n} = \lim_{N\to\infty} \sum_{k=1}^N \lim_{n\to\infty} \frac{k}{n} $$ and tried to synchronize $N$ and $n$ somehow.

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