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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-A-n-




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

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