Question Number 57194 by maxmathsup by imad last updated on 31/Mar/19
![let A_n =∫_n ^n (([(√(x+1))]−[(√x)])/x) dx with n natural integr and n≥1 1) find A_n interms of n 2)find nature of the serie Σ A_n](Q57194.png)
$${let}\:\:{A}_{{n}} =\int_{{n}} ^{{n}} \:\frac{\left[\sqrt{{x}+\mathrm{1}}\right]−\left[\sqrt{{x}}\right]}{{x}}\:{dx}\:\:\:{with}\:{n}\:{natural}\:{integr}\:{and}\:{n}\geqslant\mathrm{1} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{A}_{{n}} \:{interms}\:{of}\:{n} \\ $$$$\left.\mathrm{2}\right){find}\:{nature}\:{of}\:{the}\:{serie}\:\Sigma\:{A}_{{n}} \\ $$
Commented by maxmathsup by imad last updated on 31/Mar/19
![A_n =∫_1 ^n (([(√(x+1))]−[(√x)])/x)dx](Q57205.png)
$${A}_{{n}} =\int_{\mathrm{1}} ^{{n}} \:\frac{\left[\sqrt{{x}+\mathrm{1}}\right]−\left[\sqrt{{x}}\right]}{{x}}{dx} \\ $$