# Checking convergence of a sequence [duplicate]

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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 StackExchange.ready(function() { if (StackExchange.options.isMobile) return; $('.dupe-hammer-message-hover:not(.hover-bound)').each(function() { var$hover = $(this).addClass('hover-bound'),$msg = $hover.siblings('.dupe-hammer-message');$hover.hover( function() { $hover.showInfoMessage('', { messageElement:$msg.clone().show(), transient: false, position: { my: 'bottom left', at: 'top center', offsetTop: -7 }, dismissable: false, relativeToBody: true }); }, function() { StackExchange.helpers.removeMessages(); } ); }); }); May 30 at 15:35

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