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sinh-1-1-sinh-1-1-2-2-sinh-1-1-3-2-sinh-1-1-4-2-




Question Number 123460 by Dwaipayan Shikari last updated on 25/Nov/20
sinh^(−1) (1)+sinh^(−1) ((1/2^2 ))+sinh^(−1) ((1/3^2 ))+sinh^(−1) ((1/4^2 ))+....
$${sinh}^{−\mathrm{1}} \left(\mathrm{1}\right)+{sinh}^{−\mathrm{1}} \left(\frac{\mathrm{1}}{\mathrm{2}^{\mathrm{2}} }\right)+{sinh}^{−\mathrm{1}} \left(\frac{\mathrm{1}}{\mathrm{3}^{\mathrm{2}} }\right)+{sinh}^{−\mathrm{1}} \left(\frac{\mathrm{1}}{\mathrm{4}^{\mathrm{2}} }\right)+…. \\ $$$$ \\ $$
Commented by Dwaipayan Shikari last updated on 25/Nov/20
sinh^(−1) x=t  ⇒x=sinht⇒x=((e^t −e^(−t) )/2)⇒e^t =x±(√(x^2 +1)) ⇒t=log(x±(√(x^2 +1)))  sinh^(−1) x=log(x±(√(x^2 +1)))  sinh^(−1) (1)+sinh^(−1) ((1/2^2 ))+sinh^(−1) ((1/3^2 ))+...=log((1+(√2))(((1+(√(17)))/4))(((1+(√(82)))/9))...)
$${sinh}^{−\mathrm{1}} {x}={t} \\ $$$$\Rightarrow{x}={sinht}\Rightarrow{x}=\frac{{e}^{{t}} −{e}^{−{t}} }{\mathrm{2}}\Rightarrow{e}^{{t}} ={x}\pm\sqrt{{x}^{\mathrm{2}} +\mathrm{1}}\:\Rightarrow{t}={log}\left({x}\pm\sqrt{{x}^{\mathrm{2}} +\mathrm{1}}\right) \\ $$$${sinh}^{−\mathrm{1}} {x}={log}\left({x}\pm\sqrt{{x}^{\mathrm{2}} +\mathrm{1}}\right) \\ $$$${sinh}^{−\mathrm{1}} \left(\mathrm{1}\right)+{sinh}^{−\mathrm{1}} \left(\frac{\mathrm{1}}{\mathrm{2}^{\mathrm{2}} }\right)+{sinh}^{−\mathrm{1}} \left(\frac{\mathrm{1}}{\mathrm{3}^{\mathrm{2}} }\right)+…={log}\left(\left(\mathrm{1}+\sqrt{\mathrm{2}}\right)\left(\frac{\mathrm{1}+\sqrt{\mathrm{17}}}{\mathrm{4}}\right)\left(\frac{\mathrm{1}+\sqrt{\mathrm{82}}}{\mathrm{9}}\right)…\right) \\ $$

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