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1-6-p-logp-p-2-1-log1-pi-2-log2-4pi-2-log3-9pi-2-p-prime-

Question Number 128236 by Dwaipayan Shikari last updated on 05/Jan/21 $$\frac{\mathrm{1}}{\mathrm{6}}\underset{{p}} {\overset{\infty} {\sum}}\frac{{logp}}{{p}^{\mathrm{2}} −\mathrm{1}}=\frac{{log}\mathrm{1}}{\pi^{\mathrm{2}} }+\frac{{log}\mathrm{2}}{\mathrm{4}\pi^{\mathrm{2}} }+\frac{{log}\mathrm{3}}{\mathrm{9}\pi^{\mathrm{2}} }+…\:\:\:\left({p}={prime}\right) \\ $$ Commented by Dwaipayan Shikari last updated…

4-99-7-999-11-999999-

Question Number 128126 by Agnibhoo last updated on 04/Jan/21 $$\:\frac{\mathrm{4}}{\mathrm{99}}\:+\:\frac{\mathrm{7}}{\mathrm{999}}\:+\:\frac{\mathrm{11}}{\mathrm{999999}}\:=\:? \\ $$ Answered by Geovanek last updated on 04/Jan/21 $$\frac{\mathrm{4}}{\mathrm{99}}\:+\:\frac{\mathrm{7}}{\mathrm{99}}\:+\:\frac{\mathrm{11}}{\mathrm{999999}}\:=\:{X} \\ $$$$\mathrm{We}\:\mathrm{can}\:\mathrm{see}\:\mathrm{that} \\ $$$$\frac{\mathrm{999999}}{\mathrm{99}}\:=\:\mathrm{10101}\:\:\:\boldsymbol{\mathrm{AND}} \\…

1-1-16-5-2-16-2-2-5-2-9-2-16-3-3-5-2-9-2-13-2-16-4-4-pi-2-3-4-F-1-1-4-1-4-1-1-Prove-The-above-relation-Where-F-1-n-0-n-

Question Number 128122 by Dwaipayan Shikari last updated on 04/Jan/21 $$\mathrm{1}+\frac{\mathrm{1}}{\mathrm{16}}+\frac{\mathrm{5}^{\mathrm{2}} }{\mathrm{16}^{\mathrm{2}} .\mathrm{2}!}+\frac{\mathrm{5}^{\mathrm{2}} .\mathrm{9}^{\mathrm{2}} }{\mathrm{16}^{\mathrm{3}} .\mathrm{3}!}+\frac{\mathrm{5}^{\mathrm{2}} .\mathrm{9}^{\mathrm{2}} .\mathrm{13}^{\mathrm{2}} }{\mathrm{16}^{\mathrm{4}} .\mathrm{4}!}+…=\frac{\sqrt{\pi}}{\Gamma^{\mathrm{2}} \left(\frac{\mathrm{3}}{\mathrm{4}}\right)}={F}_{\mathrm{1}} \left(\frac{\mathrm{1}}{\mathrm{4}},\frac{\mathrm{1}}{\mathrm{4}},\mathrm{1};\mathrm{1}\right) \\ $$$${Prove}\:{The}\:{above}\:{relation} \\…

1-2-3-4-100-

Question Number 128112 by AgnibhoMukhopadhyay last updated on 04/Jan/21 $$\:\mathrm{1}\:+\:\mathrm{2}\:+\:\mathrm{3}\:+\:\mathrm{4}\:+\:…..\:+\:\mathrm{100}\:=\:? \\ $$ Answered by Olaf last updated on 04/Jan/21 $$\mathrm{A}\:\left({n}+\mathrm{1}\right)×\left({n}+\mathrm{1}\right)\:\mathrm{squared}\:\mathrm{chess}\:\mathrm{board} \\ $$$$\mathrm{contains}\:\left({n}+\mathrm{1}\right)^{\mathrm{2}} \:\mathrm{squares}. \\ $$$$…

Question-128093

Question Number 128093 by BHOOPENDRA last updated on 04/Jan/21 Answered by Dwaipayan Shikari last updated on 04/Jan/21 $$\mathscr{L}\left({e}^{\mathrm{2}{t}} +\mathrm{4}{t}^{\mathrm{3}} −\mathrm{2}{sin}\mathrm{3}{t}+\mathrm{3}{cos}\mathrm{3}{t}\right) \\ $$$$=\int_{\mathrm{0}} ^{\infty} {e}^{\mathrm{2}{t}−{st}} +\int_{\mathrm{0}}…

8-i-3-2i-If-the-expression-above-is-rewritten-in-the-form-a-bi-where-a-and-b-are-real-numbers-what-is-the-value-of-a-A-2-B-8-3-C-3-D-11-3

Question Number 128083 by AgnibhoMukhopadhyay last updated on 04/Jan/21 $$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\frac{\mathrm{8}\:−\:{i}}{\mathrm{3}\:−\:\mathrm{2}{i}} \\ $$$$\:\mathrm{If}\:\mathrm{the}\:\mathrm{expression}\:\mathrm{above}\:\mathrm{is}\:\mathrm{rewritten}\: \\ $$$$\:\mathrm{in}\:\mathrm{the}\:\mathrm{form}\:{a}\:+\:{bi},\:\mathrm{where}\:{a}\:\mathrm{and}\:{b}\:\mathrm{are} \\ $$$$\:\mathrm{real}\:\mathrm{numbers},\:\mathrm{what}\:\mathrm{is}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:{a}? \\ $$$$\:\mathrm{A}.\:\mathrm{2} \\ $$$$\:\mathrm{B}.\:\frac{\mathrm{8}}{\mathrm{3}} \\ $$$$\:\mathrm{C}.\:\mathrm{3} \\ $$$$\:\mathrm{D}.\:\frac{\mathrm{11}}{\mathrm{3}} \\…