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For $n \geq 1$, is the sequence $(x_n)_{n=1}^{\infty}$ where: $$x_n=1+\frac{1}{\sqrt{2}}+...+\frac{1}{\sqrt{n}}-2\sqrt{n}$$ convergent?

I started with $x_{n+1}-x_{n}=\frac{\sqrt{n(n+1)}-(2n+1)}{\sqrt{n+1}}\leq 0$ since geometric mean does not exceed algebraic mean, thus decreasing, but what about convergence?


marked as duplicate by José Carlos Santos real-analysis May 30 at 15:35

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