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Author: Tinku Tara

1-cos-x-cos-3x-1-cos-x-cos-5x-1-cos-x-cos-7x-1-cos-x-cos-11x-

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Question Number 208130 by efronzo1 last updated on 06/Jun/24 $$\:\:\frac{\mathrm{1}}{\mathrm{cos}\:\mathrm{x}−\mathrm{cos}\:\mathrm{3x}}\:+\:\frac{\mathrm{1}}{\mathrm{cos}\:\mathrm{x}−\mathrm{cos}\:\mathrm{5x}}\:+ \\ $$$$\:\frac{\mathrm{1}}{\mathrm{cos}\:\mathrm{x}−\mathrm{cos}\:\mathrm{7x}}\:+\:\frac{\mathrm{1}}{\mathrm{cos}\:\mathrm{x}−\mathrm{cos}\:\mathrm{11x}}=?\: \\ $$ Answered by Ghisom last updated on 06/Jun/24 $$\mathrm{let}\:{c}_{{n}} =\mathrm{cos}\:{nx} \\ $$$$=−\frac{\mathrm{2}{c}_{\mathrm{10}}…

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Find-1-4-1-8-3-64-4-256-

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Question Number 208105 by hardmath last updated on 05/Jun/24 $$\mathrm{Find}: \\ $$$$\frac{\mathrm{1}}{\mathrm{4}}\:\:+\:\:\frac{\mathrm{1}}{\mathrm{8}}\:\:+\:\:\frac{\mathrm{3}}{\mathrm{64}}\:\:+\:\:\frac{\mathrm{4}}{\mathrm{256}}\:\:+\:\:…\:\:+\:\:=\:\:? \\ $$ Commented by Frix last updated on 05/Jun/24 $$=\frac{\mathrm{1}}{\mathrm{4}}+\frac{\mathrm{2}}{\mathrm{16}}+\frac{\mathrm{3}}{\mathrm{64}}+\frac{\mathrm{4}}{\mathrm{256}}+…= \\ $$$$=\underset{{k}=\mathrm{1}} {\overset{\infty}…

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Find-inf-m-n-m-n-N-m-lt-n-2-

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Question Number 208103 by depressiveshrek last updated on 05/Jun/24 $$\mathrm{Find}\:\mathrm{inf}\left\{\frac{{m}}{{n}}\:\mid\:{m},\:{n}\:\in\:\mathbb{N},\:{m}<{n}−\mathrm{2}\right\} \\ $$ Answered by A5T last updated on 05/Jun/24 $${Since}\:{m},{n}\in\mathbb{N},\:\frac{{m}}{{n}}>\mathrm{0},{so}\:\mathrm{0}\:{is}\:{a}\:{lower}\:{bound}. \\ $$$$\frac{{m}}{{n}}\geqslant\frac{\mathrm{1}}{{n}}\:{and}\:\frac{\mathrm{1}}{{n}}\:{is}\:{in}\:{the}\:{set}. \\ $$$${For}\:{any}\:\epsilon>\mathrm{0},\:{one}\:{can}\:{always}\:{find}\:{n}\:{such}\:{that} \\…

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