Convergence of a sequence, $a_n=\sum_1^nn/(n^2+k)$

real-analysis sequences-and-series summation limits

Solution 1:

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

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

So $a_n\rightarrow 1$.

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