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Prove-k-1-k-sinh-pik-1-4-4-8pi-2-32pi-3-

Question Number 222276 by Nicholas666 last updated on 27/Jun/25 $$ \\ $$$$\:\mathrm{Prove};\:\underset{{k}\:=\:\mathrm{1}} {\overset{\infty} {\sum}}\:\frac{{k}}{\mathrm{sinh}\left(\pi{k}\right)}\:=\:\frac{\Gamma\left(\frac{\mathrm{1}}{\mathrm{4}}\right)^{\mathrm{4}} −\:\mathrm{8}\pi^{\mathrm{2}} }{\mathrm{32}\pi^{\mathrm{3}} }\:\:\:\:\:\:\:\:\:\:\:\:\:\: \\ $$$$ \\ $$ Terms of Service Privacy…

Question-221814

Question Number 221814 by Tawa11 last updated on 10/Jun/25 Answered by wewji12 last updated on 11/Jun/25 $$\mathrm{log}_{{a}} \left({b}\right)=\frac{\mathrm{log}_{{b}} \left({c}\right)}{\mathrm{log}_{{a}} \left({c}\right)} \\ $$$$\frac{\mathrm{ln}\left(\mathrm{2}\right)}{\mathrm{ln}\left({x}+\mathrm{1}\right)}+\frac{\mathrm{ln}\left(\mathrm{3}\right)}{\mathrm{ln}\left({x}+\mathrm{2}\right)}+\frac{\mathrm{ln}\left(\mathrm{4}\right)}{\mathrm{ln}\left({x}+\mathrm{3}\right)}=\mathrm{3} \\ $$$$\mathrm{if}\:{x}=\mathrm{1}\: \\…