Question Number 185023 by aba last updated on 17/Jan/23 $$\mathrm{provet}\:\mathrm{that}\::\:\underset{\mathrm{k}=\mathrm{1}} {\overset{\mathrm{n}} {\sum}}\frac{\mathrm{1}}{\mathrm{n}+\mathrm{k}}>\mathrm{ln}\left(\frac{\mathrm{2n}+\mathrm{1}}{\mathrm{n}+\mathrm{1}}\right) \\ $$ Answered by witcher3 last updated on 18/Jan/23 $$\mathrm{ln}\left(\frac{\mathrm{2n}+\mathrm{1}}{\mathrm{n}+\mathrm{1}}\right)=\int_{\mathrm{n}+\mathrm{1}} ^{\mathrm{2n}+\mathrm{1}} \frac{\mathrm{1}}{\mathrm{x}}\mathrm{dx}=\underset{\mathrm{k}=\mathrm{1}} {\overset{\mathrm{n}}…
Question Number 185012 by aba last updated on 15/Jan/23 $$\underset{\mathrm{k}=\mathrm{0}} {\overset{\mathrm{n}} {\sum}}\mathrm{sin}\left(\mathrm{kx}\right)=\underset{\mathrm{k}=\mathrm{0}} {\overset{\mathrm{n}} {\sum}}\mathrm{Im}\left(\mathrm{e}^{\mathrm{ikx}} \right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\mathrm{Im}\left(\underset{\mathrm{k}=\mathrm{0}} {\overset{\mathrm{n}} {\sum}}\left(\mathrm{e}^{\mathrm{ix}} \right)^{\mathrm{k}} \right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\mathrm{Im}\left(\frac{\mathrm{1}−\mathrm{e}^{\mathrm{i}\left(\mathrm{n}+\mathrm{1}\right)\mathrm{x}} }{\mathrm{1}−\mathrm{e}^{\mathrm{ix}} }\right)…
Question Number 119474 by Aviga last updated on 24/Oct/20 $$ \\ $$$$\boldsymbol{{P}\mathrm{lease}}\:\boldsymbol{\mathrm{help}} \\ $$$$\mathrm{create}\:\mathrm{an}\:\mathrm{algorithm}\:\mathrm{when}\:\mathrm{the}\: \\ $$$$\mathrm{userviews}\:\mathrm{in}\:\mathrm{website}\:\mathrm{around}\:\mathrm{0}− \\ $$$$\mathrm{11}:\mathrm{59}\:\mathrm{it}\:\mathrm{will}\:\mathrm{alert}\:“\mathrm{Good}\:\mathrm{Morning}''\: \\ $$$$\mathrm{and}\:\mathrm{from}\:\mathrm{12}−\:\mathrm{15}:\mathrm{59}\:\mathrm{it}\:\mathrm{will}\:\mathrm{alert}\: \\ $$$$“\mathrm{Goodafternoon}''\:\mathrm{and}\:\mathrm{lastly}\:\mathrm{from}\: \\ $$$$\mathrm{16}−\:\mathrm{23}:\mathrm{59}\:\mathrm{it}\:\mathrm{will}\:\mathrm{alert}\:“\mathrm{Good}\:\mathrm{evening}''. \\…
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Question Number 185008 by paul2222 last updated on 15/Jan/23 Answered by witcher3 last updated on 26/Jan/23 $$\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\mathrm{x}}{\mathrm{1}+\mathrm{x}^{\mathrm{4}} }\mathrm{ln}\left(\mathrm{1}−\mathrm{x}\right)\mathrm{dx}=\mathrm{A},\mathrm{B}=\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\mathrm{x}}{\mathrm{1}+\mathrm{x}^{\mathrm{4}} }\mathrm{ln}\left(\mathrm{1}+\mathrm{x}\right) \\ $$$$\mathrm{A}+\mathrm{B}=\int_{\mathrm{0}}…
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Question Number 119473 by help last updated on 24/Oct/20 Commented by 1549442205PVT last updated on 25/Oct/20 $$\mathrm{I}\:\mathrm{think}\:\mathrm{that}\:\mathrm{question}\:\mathrm{isn}'\mathrm{t}\:\mathrm{clearly}.\mathrm{You} \\ $$$$\mathrm{need}\:\mathrm{must}\:\mathrm{add}\:\mathrm{that}\:\mathrm{above}\:\mathrm{equality}\:\mathrm{be} \\ $$$$\mathrm{an}\:\mathrm{identity}\:\mathrm{that}\:\mathrm{means}\:\mathrm{it}\:\mathrm{is}\:\mathrm{true}\:\mathrm{for} \\ $$$$\forall\mathrm{x}\neq−\mathrm{1}/\mathrm{2}\:\mathrm{othewise}\:\mathrm{we}\:\mathrm{can}\:\mathrm{find}\:\mathrm{infinite} \\ $$$$\mathrm{many}\:\mathrm{values}\:\mathrm{of}\:\mathrm{m}\:\mathrm{satisfying}\:\mathrm{the}\:\mathrm{condition}…
Question Number 184994 by Ml last updated on 15/Jan/23 $$\sqrt{\mathrm{1}+\sqrt{\mathrm{2}+\sqrt{\mathrm{3}+\sqrt{\mathrm{4}+….+\sqrt{\mathrm{70}}}}}}=? \\ $$ Commented by Frix last updated on 15/Jan/23 $$\mathrm{Use}\:\mathrm{a}\:\mathrm{calculator}. \\ $$$$\mathrm{Do}\:\mathrm{you}\:\mathrm{think}\:\sqrt{…+\sqrt{\mathrm{70}}}\:\mathrm{is}\:\mathrm{more}\:\mathrm{special}\:\mathrm{than} \\ $$$$\sqrt{…+\sqrt{\mathrm{71}}}\:\mathrm{or}\:\sqrt{…+\sqrt{\mathrm{69}}}?\:\mathrm{In}\:\mathrm{fact}\:\mathrm{the}\:\mathrm{only}\:\mathrm{one} \\…
Question Number 184992 by SANOGO last updated on 15/Jan/23 $${prove}\:{that}: \\ $$$$\int_{{o}} ^{\mathrm{1}} \frac{{sint}}{{e}^{{t}} −\mathrm{1}}=\underset{{n}={o}} {\sum}\:\frac{\mathrm{1}}{{n}^{\mathrm{2}} +\mathrm{1}} \\ $$ Answered by ARUNG_Brandon_MBU last updated on…