Question and Answers Forum

All Questions      Topic List

Coordinate Geometry Questions

Previous in All Question      Next in All Question      

Previous in Coordinate Geometry      Next in Coordinate Geometry      

Question Number 197740 by cortano12 last updated on 27/Sep/23

Answered by mr W last updated on 27/Sep/23

lim_(n→∞) Σ_(k=1) ^n ((k(√k))/n)  =lim_(n→∞) ((n(√n))/n)Σ_(k=1) ^n ((k(√k))/(n(√n)))  =(lim_(n→∞) n(√n))[lim_(n→∞) (1/n)Σ_(k=1) ^n ((k/n)(√(k/n)))]  =(lim_(n→∞) n(√n))[∫_0 ^1 x(√x)dx]  =(lim_(n→∞) n(√n))×(2/5)  =∞

$$\underset{{n}\rightarrow\infty} {\mathrm{lim}}\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}\frac{{k}\sqrt{{k}}}{{n}} \\ $$$$=\underset{{n}\rightarrow\infty} {\mathrm{lim}}\frac{{n}\sqrt{{n}}}{{n}}\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}\frac{{k}\sqrt{{k}}}{{n}\sqrt{{n}}} \\ $$$$=\left(\underset{{n}\rightarrow\infty} {\mathrm{lim}}{n}\sqrt{{n}}\right)\left[\underset{{n}\rightarrow\infty} {\mathrm{lim}}\frac{\mathrm{1}}{{n}}\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}\left(\frac{{k}}{{n}}\sqrt{\frac{{k}}{{n}}}\right)\right] \\ $$$$=\left(\underset{{n}\rightarrow\infty} {\mathrm{lim}}{n}\sqrt{{n}}\right)\left[\int_{\mathrm{0}} ^{\mathrm{1}} {x}\sqrt{{x}}{dx}\right] \\ $$$$=\left(\underset{{n}\rightarrow\infty} {\mathrm{lim}}{n}\sqrt{{n}}\right)×\frac{\mathrm{2}}{\mathrm{5}} \\ $$$$=\infty \\ $$

Answered by witcher3 last updated on 27/Sep/23

Σ_(k=1) ^n k(√k)≥n(√n)  ((Σk(√k))/n)≥(√n)→∞

$$\underset{\mathrm{k}=\mathrm{1}} {\overset{\mathrm{n}} {\sum}}\mathrm{k}\sqrt{\mathrm{k}}\geqslant\mathrm{n}\sqrt{\mathrm{n}} \\ $$$$\frac{\Sigma\mathrm{k}\sqrt{\mathrm{k}}}{\mathrm{n}}\geqslant\sqrt{\mathrm{n}}\rightarrow\infty \\ $$

Answered by Frix last updated on 27/Sep/23

((1+2^? (√2)+3(√3)+...+n(√n))/n)=s+(√n)  s>0∧lim_(n→∞)  (√n) =∞

$$\frac{\mathrm{1}+\overset{?} {\mathrm{2}}\sqrt{\mathrm{2}}+\mathrm{3}\sqrt{\mathrm{3}}+...+{n}\sqrt{{n}}}{{n}}={s}+\sqrt{{n}} \\ $$$${s}>\mathrm{0}\wedge\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:\sqrt{{n}}\:=\infty \\ $$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com