Prove that

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

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

Cooper Doyle
2022-07-07
Answered

Prove that

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

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

You can still ask an expert for help

Valeria Wolfe

Answered 2022-07-08
Author has **11** answers

you have

$\frac{1}{n}-\frac{1}{n+1}=\frac{1}{n(n+1)}$

and

$0\le \frac{1}{n(n+1)}\le \frac{1}{{n}^{2}}$

then because

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

we have $\underset{n\to +\mathrm{\infty}}{lim}(\frac{1}{n}-\frac{1}{n+1})=0$

$\frac{1}{n}-\frac{1}{n+1}=\frac{1}{n(n+1)}$

and

$0\le \frac{1}{n(n+1)}\le \frac{1}{{n}^{2}}$

then because

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

we have $\underset{n\to +\mathrm{\infty}}{lim}(\frac{1}{n}-\frac{1}{n+1})=0$

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