Ayaana Buck

## Answered question

2021-08-19

Try to find the limits of the following sequences.
1)$\left(1-\frac{1}{2}\right),\left(\frac{1}{2}-\frac{1}{3}\right),\left(\frac{1}{3}-\frac{1}{4}\right),\left(\frac{1}{4}-\frac{1}{5}\right),\dots$
2)$\left(\sqrt{2}-\sqrt{3}\right),\left(\sqrt{3}-\sqrt{4}\right),\left(\sqrt{4}-\sqrt{5}\right),\dots$

### Answer & Explanation

Jaylen Fountain

Skilled2021-08-20Added 169 answers

1)
${\left\{\frac{1}{n}-\frac{1}{n+1}\right\}}_{n-1}^{\mathrm{\infty }}$
$\underset{n\to \mathrm{\infty }}{lim}\left(\frac{1}{n}-\frac{1}{n+1}\right)=\underset{n\to \mathrm{\infty }}{lim}\frac{1}{n\left(n+1\right)}=0$
$\underset{n\to \mathrm{\infty }}{lim}\left(\frac{1}{n}-\frac{1}{n+1}\right)=0$

2)
$\left(\sqrt{n+2}-\sqrt{n}+3{\right)}_{n=1}^{\mathrm{\infty }}$
$\underset{n\to \mathrm{\infty }}{lim}\sqrt{n+2}-\sqrt{n}+3=\underset{n\to \mathrm{\infty }}{lim}\frac{\left(\sqrt{n}+2-\sqrt{n}+3\right)\sqrt{n}+2+\sqrt{n}+3}{\sqrt{n+2}+\sqrt{n}+3}$
$=\underset{n\to \mathrm{\infty }}{lim}\frac{-1}{\sqrt{n+2}+\sqrt{n}+3}=0$
$\underset{n\to \mathrm{\infty }}{lim}\sqrt{n}+2-\sqrt{n+3}=0$

Jeffrey Jordon

Expert2021-10-23Added 2605 answers

Answer is given below (on video)

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