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Question Number 185337 by Shrinava last updated on 20/Jan/23

Answered by qaz last updated on 20/Jan/23

Σ_(k=2) ^n ((k+(√k))/k)∼∫_2 ^n ((x+(√x))/x)dx∼n Σ_(k=2) ^n ((k+(k)^(1/3) )/k)∼∫_2 ^n ((x+(x)^(1/3) )/k)dx∼n Σ_(k=2) ^n ((k+(k)^(1/5) )/k)∼∫_2 ^n ((x+(x)^(1/5) )/x)dx∼n Σ_(k=2) ^n k^2 ∼(1/3)n^3 ⇒Ω=3

$$\underset{{k}=\mathrm{2}} {\overset{{n}} {\sum}}\frac{{k}+\sqrt{{k}}}{{k}}\sim\int_{\mathrm{2}} ^{{n}} \frac{{x}+\sqrt{{x}}}{{x}}{dx}\sim{n} \\ $$$$\underset{{k}=\mathrm{2}} {\overset{{n}} {\sum}}\frac{{k}+\sqrt[{\mathrm{3}}]{{k}}}{{k}}\sim\int_{\mathrm{2}} ^{{n}} \frac{{x}+\sqrt[{\mathrm{3}}]{{x}}}{{k}}{dx}\sim{n} \\ $$$$\underset{{k}=\mathrm{2}} {\overset{{n}} {\sum}}\frac{{k}+\sqrt[{\mathrm{5}}]{{k}}}{{k}}\sim\int_{\mathrm{2}} ^{{n}} \frac{{x}+\sqrt[{\mathrm{5}}]{{x}}}{{x}}{dx}\sim{n} \\ $$$$\underset{{k}=\mathrm{2}} {\overset{{n}} {\sum}}{k}^{\mathrm{2}} \sim\frac{\mathrm{1}}{\mathrm{3}}{n}^{\mathrm{3}} \\ $$$$\Rightarrow\Omega=\mathrm{3} \\ $$

Commented by Shrinava last updated on 23/Jan/23

Thank you dear professor, is there any other solution?

$$\mathrm{Thank}\:\mathrm{you}\:\mathrm{dear}\:\mathrm{professor}, \\ $$$$\mathrm{is}\:\mathrm{there}\:\mathrm{any}\:\mathrm{other}\:\mathrm{solution}? \\ $$

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