Limit of $\sum \limits_{k=1}^{n} \frac{2^kn+2n^2+k}{2^{k+1}n^2+2^kk}$ when $n\to\infty$ I have to show the convergence of the series
$$\lim\limits_{n \to \infty}a_n=\sum \limits_{k=1}^{n} \frac{2^kn+2n^2+k}{2^{k+1}n^2+2^kk}.$$ I am quite sure that the limit is 1.5. I wanted to show this by estimation for another series smaller than $a_n$ and a further one that is bigger than $\lim\limits_{n \to \infty}a_n$.
I have been trying to find this estimation for hours without success. Can someone help me, please?

break it up.
$a_n=\sum \limits_{k=1}^{n} \frac{2^kn}{2^{k+1}n^2+2^kk} + \sum\limits_{k=1}^{n} \frac{2n^2}{2^{k+1}n^2+2^kk}+\sum\limits_{k=1}^{n} \frac{k}{2^{k+1}n^2+2^kk}.$
Since every element of each of those series is greater than zero, 
the series for $a_n$ converges iff all three of the above converge, and diverges if any of the above diverge.
$\sum \limits_{k=1}^{n} \frac{2^kn}{2^{k+1}n^2+2^kk} <\sum\limits_{k=1}^{n} \frac{1}{2n}$
$\sum\limits_{k=1}^{n} \frac{2n^2}{2^{k+1}n^2+2^kk} <\sum\limits_{k=1}^{n} \frac{1}{2^{k}}$
$\sum\limits_{k=1}^{n} \frac{k}{2^{k+1}n^2+2^kk} < \sum\limits_{k=1}^{n} \frac{1}{2^{k}}$
