Sequence Limit. Convergence Determine whether the sequence $a_n=\frac {1^1} {n^2}+\frac {2^1} {n^2}+\cdots+ \frac {n^1} {n^2}$ converges or diverges. If it converges, find the limit. 
Note that, $\sum_{i=1}^k i=\frac {k(k+1)} 2$.
I'm assuming it is a $p$ series and because $p>1,$ it must converge. But, I do not know how to find the limit of it.
 A: It looks like
$$
a_n = \frac{1}{n^2} \sum_{i = 1}^{n} i
 = \frac{1}{n^2} \frac{n(n + 1)}2
 = \frac{n^2 + n}{2n^2}
= \frac{1}{2} + \frac{1}{2n}
$$
Which means that $\lim_{n \to \infty} a_n = 1/2$ .
A: So, you have $a_n=\sum_{i=1}^n\frac{i}{n^2}=\frac{\sum_{i=1}^ni}{n^2}=\frac{n(n+1)}{2n^2}=\frac{n+1}{2n}$. So, the limit $n\to\infty$ is $\frac{1}{2}$.
A: This is NOT a question about a sum of an infinite series.
An infinite series looks like this:
$$
\sum_{k=1}^\infty a_k = \lim_{n\to\infty} \big( a_1 + \cdots + a_n\big). 
$$
Notice what happens each time $n$ is incremented by one unit, for example when $n$ goes from $100$ to $101$:
\begin{align}
a_1 + \cdots + a_{100} & \\
a_1 + \cdots + a_{100} & {} + a_{101}
\end{align}
A new term is added, but the terms that were already there, $a_1,\ldots,a_{100}$ do not change.
However, in the problem you have posed, those terms change every time $n$ is incremented.
$$
\frac{ 1+ 2 + 3 + \cdots + n } {n^2} = \frac{\left( \frac{n(n+1)} 2 \right)}{n^2}
$$
and so on.
