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 March 9th, 2013, 06:17 AM #1 Newbie   Joined: Mar 2013 Posts: 6 Thanks: 0 Convergence of a sequence How do I prove n/(2n + sqrt(n)) converges using an epsilon delta proof.
 March 9th, 2013, 10:21 AM #2 Math Team   Joined: Nov 2010 From: Greece, Thessaloniki Posts: 1,990 Thanks: 133 Math Focus: pre pre pre pre pre pre pre pre pre pre pre pre calculus Re: Convergence of a sequence [color=#000000]$a_{n}=\frac{n}{2n+\sqrt{n}}=\frac{\frac{n}{n}}{\fr ac{2n}{n}+\frac{\sqrt{n}}{n}}=\frac{1}{2+\frac{\sq rt{n}}{n}}$ We will prove that $\frac{\sqrt{n}}{n}\to 0$, for every ?>0, we will find a suitable $n_{0}=n(\epsilon)$ such that $\left|\frac{\sqrt{n}}{n}\right|<\epsilon$. From the last inequality we get $\left|\frac{\sqrt{n}}{n}\right|<\epsilon\Rightarro w \left|\frac{1}{\sqrt{n}}\right|<\epsilon\Rightarro w n>\frac{1}{\epsilon^2}$. So we proved that for all ?>0 and $n>\frac{1}{\epsilon^2}$, $|a_{n}-\frac{1}{2}|<\epsilon$, meaning that $a_{n}\to\frac{1}{2}$.[/color]
 March 9th, 2013, 10:33 AM #3 Newbie   Joined: Mar 2013 Posts: 6 Thanks: 0 Re: Convergence of a sequence Thank you. That helped me understand.

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