Convergence of a sequence (possibly Riemann sum)

Let $a_1, a_2, a_3, . . . , a_n$ be the sequence defined by
$$
a_n = 2\sqrt{n}-\sum_{k=1}^{n}\frac{1}{\sqrt{k}} = 2\sqrt{n} - \frac{1}{\sqrt{1}}-\frac{1}{\sqrt{2}}-...-\frac{1}{\sqrt{n}}
$$

show that the sequence $a_n$ is convergent to some limit L, and that $1

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I tried looking at this as a Riemann sum. However, I failed to covert it to that. Any hints on that or alternate solution? Thanks