tripplek7i

2022-01-24

Prove that $\underset{n\to \mathrm{\infty}}{lim}\frac{1}{{n}^{2}}=0$

marzembreax

Beginner2022-01-25Added 13 answers

Use the definition of convergent

$\underset{n\to \mathrm{\infty}}{lim}\frac{1}{{n}^{2}}=0$

Let$\u03f5$ be any positive number

We have to show that$|\frac{1}{{n}^{2}}-0|<\u03f5$

Now, take$N\in N$ , $\frac{1}{\u03f5}<N$ , then

$\mathrm{\forall}n>N(ie.\frac{1}{n}<\frac{1}{N})$

and$|\frac{1}{{n}^{2}}-0|=\left|\frac{1}{{n}^{2}}\right|=\frac{1}{{n}^{2}}<\frac{1}{N}<\u03f5$

($\frac{1}{{n}^{2}}$ is always positive)

Hence,$|\frac{1}{{n}^{2}}-0|<\u03f5$

$\underset{n\to \mathrm{\infty}}{lim}\frac{1}{{n}^{2}}=0$

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