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Let $ a_{n} = \sum_{k=1}^{n} \frac{n}{n^{2}+k}$ . I would like to know whether the given sequence converges.

I see that,

$ a_{n} = \sum_{k=1}^{n} \frac{n}{n^{2}+k}= \sum_{k=1}^{n} \frac{1}{n+\frac{k}{n}}.$ When $n$ gets sufficiently large the contribution by the $ \frac{k}{n} $ term is diminishing and $ a_{n} < \sum_{k=1}^{n} \frac{1}{n} = 1 $.

Thank you.

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    Thank you all .I found your discussions to be very useful!2011-11-01

1 Answers 1

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$\frac{1}{n+1}\leq \frac{1}{n+\frac{k}n} \leq \frac{1}{n}$

So $\frac{n}{n+1} \leq a_n \leq 1$

So $a_n\rightarrow 1$.