Question Number 159723 by stelor last updated on 20/Nov/21
$$\frac{\mathrm{1}}{{k}+\mathrm{1}}\leqslant\int_{{k}} ^{{k}+\mathrm{1}} \left(\frac{\mathrm{1}}{{x}}\right){dx}\leqslant\frac{\mathrm{1}}{{k}}\:\:\:\:\:\:\:\:\: \\ $$$$\:{please}\:{show}\:{it}\:{with}\:{k}\in\aleph−\left(\mathrm{0}\right) \\ $$$$ \\ $$
Commented by stelor last updated on 20/Nov/21
$$\frac{\mathrm{1}}{{k}+\mathrm{1}}\leqslant\int_{{k}} ^{{k}+\mathrm{1}} \left(\frac{\mathrm{1}}{{x}}\right){dx}\leqslant\frac{\mathrm{1}}{{k}}\:\:\:\:\:\:\:\:\: \\ $$$$\:{please}\:{show}\:{it}\:{with}\:{k}\in\aleph−\left(\mathrm{0}\right) \\ $$$$ \\ $$
Answered by puissant last updated on 20/Nov/21
$$\:\:{k}\:\leqslant\:{x}\:\leqslant\:{k}+\mathrm{1}\:\:\:\Rightarrow\:\:\frac{\mathrm{1}}{{k}+\mathrm{1}}\:\leqslant\:\frac{\mathrm{1}}{{x}}\:\leqslant\:\frac{\mathrm{1}}{{k}}\: \\ $$$$\Rightarrow\:\frac{\mathrm{1}}{{k}+\mathrm{1}}\int_{{k}} ^{{k}+\mathrm{1}} {dx}\:\leqslant\:\int_{{k}} ^{{k}+\mathrm{1}} \frac{\mathrm{1}}{{x}}\:{dx}\:\leqslant\:\frac{\mathrm{1}}{{k}}\int_{{k}} ^{{k}+\mathrm{1}} {dx} \\ $$$$\Rightarrow\:\frac{\mathrm{1}}{{k}+\mathrm{1}}\:\leqslant\:\int_{{k}} ^{{k}+\mathrm{1}} \frac{\mathrm{1}}{{x}}\:{dx}\:\leqslant\:\frac{\mathrm{1}}{{k}}... \\ $$
Commented by stelor last updated on 20/Nov/21
$${nice}. \\ $$