Menu Close

find-the-nature-of-u-n-u-n-1-2-n-n-3-




Question Number 32041 by abdo imad last updated on 18/Mar/18
find the nature of Σ u_n   /  u_n = (((√1) +(√2) +....+(√n))/n^3 ) .
$${find}\:{the}\:{nature}\:{of}\:\Sigma\:{u}_{{n}} \:\:/ \\ $$$${u}_{{n}} =\:\frac{\sqrt{\mathrm{1}}\:+\sqrt{\mathrm{2}}\:+….+\sqrt{{n}}}{{n}^{\mathrm{3}} }\:. \\ $$
Commented by abdo imad last updated on 20/Mar/18
we have u_n = (1/n^2 ) ( (1/( (√n))) Σ_(k=1) ^n  (√( (k/n))) )  =(1/(n(√n)))( (1/n) Σ_(k=1) ^n  (√(k/n)) ) but lim_(n→∞) (1/n) Σ_(k=1) ^n  (√(k/n))  = ∫_0 ^1  (√x) dx = [ (2/3) x^(3/2)  ]_0 ^1  = (2/3)  ⇒  u_n  ∼  (2/(3n(√n)))  and  the serie  Σ_(n≥1)      (2/(3n(√n))) is convergent so Σ u_(n ) converges.
$${we}\:{have}\:{u}_{{n}} =\:\frac{\mathrm{1}}{{n}^{\mathrm{2}} }\:\left(\:\frac{\mathrm{1}}{\:\sqrt{{n}}}\:\sum_{{k}=\mathrm{1}} ^{{n}} \:\sqrt{\:\frac{{k}}{{n}}}\:\right) \\ $$$$=\frac{\mathrm{1}}{{n}\sqrt{{n}}}\left(\:\frac{\mathrm{1}}{{n}}\:\sum_{{k}=\mathrm{1}} ^{{n}} \:\sqrt{\frac{{k}}{{n}}}\:\right)\:{but}\:{lim}_{{n}\rightarrow\infty} \frac{\mathrm{1}}{{n}}\:\sum_{{k}=\mathrm{1}} ^{{n}} \:\sqrt{\frac{{k}}{{n}}} \\ $$$$=\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\sqrt{{x}}\:{dx}\:=\:\left[\:\frac{\mathrm{2}}{\mathrm{3}}\:{x}^{\frac{\mathrm{3}}{\mathrm{2}}} \:\right]_{\mathrm{0}} ^{\mathrm{1}} \:=\:\frac{\mathrm{2}}{\mathrm{3}}\:\:\Rightarrow\:\:{u}_{{n}} \:\sim\:\:\frac{\mathrm{2}}{\mathrm{3}{n}\sqrt{{n}}}\:\:{and} \\ $$$${the}\:{serie}\:\:\sum_{{n}\geqslant\mathrm{1}} \:\:\:\:\:\frac{\mathrm{2}}{\mathrm{3}{n}\sqrt{{n}}}\:{is}\:{convergent}\:{so}\:\Sigma\:{u}_{{n}\:} {converges}. \\ $$

Leave a Reply

Your email address will not be published. Required fields are marked *