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consider the sequence $\{a_{n}\} _{n=1}^{\infty}$,

$$a_n= \frac{1}{n^2}+\frac{2}{n^2}+\cdots+\frac{n}{n^2} $$

(1) find $a_{1}$, $a_2$, $a_3$, $a_4$' (2) by expressing $a_n$ in closed form, calculate $\lim_{n\to\infty} a_n$

For the first question, $a_1=1$, $a_2=\frac{5}{4}$, $a_3=\frac{19}{12}$, $a_4=\frac{11}{6}$. But I don't really understand question 2, can you show me the way to do it?

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All the terms have the same denominator so you just need $1+2+ \cdots+n = n(n+1)/2$.

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Write $$ a_n = \frac{1}{n^2} \sum_{i=1}^n i = \frac{1}{n^2} \cdot \frac{n(n+1)}{2} = \frac{n+1}{2n}. $$

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