Question Number 68693 by Maclaurin Stickker last updated on 15/Sep/19
![](https://www.tinkutara.com/question/9295.png)
Commented by turbo msup by abdo last updated on 15/Sep/19
![let S_n =Σ_(k=1) ^n ((1/( (√k)))−(1/( (√(k+1))))) ⇒ S_n =1−(1/( (√2)))+(1/( (√2)))−(1/( (√3))) +...+(1/( (√n)))−(1/( (√(n+1)))) =1−(1/( (√(n+1)))) ⇒lim_(n→+∞) S_n =1](https://www.tinkutara.com/question/Q68701.png)
$${let}\:{S}_{{n}} =\sum_{{k}=\mathrm{1}} ^{{n}} \left(\frac{\mathrm{1}}{\:\sqrt{{k}}}−\frac{\mathrm{1}}{\:\sqrt{{k}+\mathrm{1}}}\right)\:\Rightarrow \\ $$$${S}_{{n}} =\mathrm{1}−\frac{\mathrm{1}}{\:\sqrt{\mathrm{2}}}+\frac{\mathrm{1}}{\:\sqrt{\mathrm{2}}}−\frac{\mathrm{1}}{\:\sqrt{\mathrm{3}}}\:+…+\frac{\mathrm{1}}{\:\sqrt{{n}}}−\frac{\mathrm{1}}{\:\sqrt{{n}+\mathrm{1}}} \\ $$$$=\mathrm{1}−\frac{\mathrm{1}}{\:\sqrt{{n}+\mathrm{1}}}\:\Rightarrow{lim}_{{n}\rightarrow+\infty} \:\:{S}_{{n}} =\mathrm{1} \\ $$
Commented by Maclaurin Stickker last updated on 15/Sep/19
![Fabulous.](https://www.tinkutara.com/question/Q68702.png)
$${Fabulous}. \\ $$