Two different results of calculation of the limit of $(2+4+\dots+2n)/n^2$ I would like to know where is the problem when i calculated those two limit with the following ways :
method (1):

$$\lim_{n\to \infty} \dfrac { 2+4+6+\cdots 2n} {n²} = \lim_{n\to \infty} \dfrac {n(n+1)}{n²}  =1 $$( using arithmitic sequence sum ).

Method (2) : now by using operation over limit we get this:
{n²}$$\lim_{n\to \infty} \dfrac { 2+4+6+\cdots 2n} {n²} = \lim_{n\to \infty} \dfrac {2}{n²}+\lim_{n\to\infty}\dfrac {4}{n²}+\cdots \lim_{n\to\infty}\dfrac {2n}{n²}  =0+0+\cdots 0=0 $$ .

Then is there some one who show me where is the problem in second method ?
Thank you for any help
You are trying
$$
\lim_{n\to\infty} \sum_{k=1}^n \frac{2k}{n^2}
"=" \sum_{k=1}^n \lim_{n\to\infty} \frac{2k}{n^2}
"=" \sum_{k=1}^n \lim_{m \to\infty} \frac{2k}{m^2}
$$
but $n$ has to be constant within the summation scope. 
For the middle term it can not stay both fixed and be subject of a limit.
Only the first and last term are well defined (and different).
in method 2 you are summing an infinite number of zeros: the result is undefined, not 0.