Question Number 68693 by Maclaurin Stickker last updated on 15/Sep/19
Commented by turbo msup by abdo last updated on 15/Sep/19
$${let}\:{S}_{{n}} =\sum_{{k}=\mathrm{1}} ^{{n}} \left(\frac{\mathrm{1}}{\sqrt{{k}}}−\frac{\mathrm{1}}{\sqrt{{k}+\mathrm{1}}}\right)\:\Rightarrow \\ $$$${S}_{{n}} =\mathrm{1}−\frac{\mathrm{1}}{\sqrt{\mathrm{2}}}+\frac{\mathrm{1}}{\sqrt{\mathrm{2}}}−\frac{\mathrm{1}}{\sqrt{\mathrm{3}}}\:+...+\frac{\mathrm{1}}{\sqrt{{n}}}−\frac{\mathrm{1}}{\sqrt{{n}+\mathrm{1}}} \\ $$$$=\mathrm{1}−\frac{\mathrm{1}}{\sqrt{{n}+\mathrm{1}}}\:\Rightarrow{lim}_{{n}\rightarrow+\infty} \:\:{S}_{{n}} =\mathrm{1} \\ $$
Commented by Maclaurin Stickker last updated on 15/Sep/19
$${Fabulous}. \\ $$