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lim-n-1-2-3-4-5-6-n-1-2-1-2-1-2-1-2-1-2-1-2-1-2-




Question Number 99793 by bachamohamed last updated on 23/Jun/20
  lim_(n→∞)  (1(2(3(4(5(6(....(n.)^(1/2) ..)^(1/2) )^(1/2) )^(1/2) )^(1/2) )^(1/2) )^(1/2) =?
$$\:\:\boldsymbol{{li}}\underset{\boldsymbol{{n}}\rightarrow\infty} {\boldsymbol{{m}}}\:\left(\mathrm{1}\left(\mathrm{2}\left(\mathrm{3}\left(\mathrm{4}\left(\mathrm{5}\left(\mathrm{6}\left(….\left(\mathrm{n}.\right)^{\frac{\mathrm{1}}{\mathrm{2}}} ..\right)^{\frac{\mathrm{1}}{\mathrm{2}}} \right)^{\frac{\mathrm{1}}{\mathrm{2}}} \right)^{\frac{\mathrm{1}}{\mathrm{2}}} \right)^{\frac{\mathrm{1}}{\mathrm{2}}} \right)^{\frac{\mathrm{1}}{\mathrm{2}}} \right)^{\frac{\mathrm{1}}{\mathrm{2}}} =?\:\:\:\right. \\ $$
Commented by Dwaipayan Shikari last updated on 23/Jun/20
(√(1(√6)))
$$\sqrt{\mathrm{1}\sqrt{\mathrm{6}}} \\ $$$$ \\ $$
Commented by bachamohamed last updated on 23/Jun/20
but   3=(√(1+2(√(1+3(√(1+4(√(1+5(√(.....))))))))))=(√(1+(2(√(1+(3(√(1+((√(1+(4(√(.....))))))))))))))...)))  3≠(√(3(√(4(√(5(√(6(√(7....))))))))))
$$\boldsymbol{{but}}\: \\ $$$$\left.\mathrm{3}=\sqrt{\mathrm{1}+\mathrm{2}\sqrt{\mathrm{1}+\mathrm{3}\sqrt{\mathrm{1}+\mathrm{4}\sqrt{\mathrm{1}+\mathrm{5}\sqrt{…..}}}}}=\sqrt{\mathrm{1}+\left(\mathrm{2}\sqrt{\left.\mathrm{1}+\left(\mathrm{3}\sqrt{\mathrm{1}+\left(\sqrt{\mathrm{1}+\left(\mathrm{4}\sqrt{\left……\right)}\right)}\right)}\right)\right)}\right)…}\right) \\ $$$$\mathrm{3}\neq\sqrt{\mathrm{3}\sqrt{\mathrm{4}\sqrt{\mathrm{5}\sqrt{\mathrm{6}\sqrt{\mathrm{7}….}}}}} \\ $$
Commented by Dwaipayan Shikari last updated on 23/Jun/20
I have come to know that my process is wrong. I am trying another method
$${I}\:{have}\:{come}\:{to}\:{know}\:{that}\:{my}\:{process}\:{is}\:{wrong}.\:{I}\:{am}\:{trying}\:{another}\:{method} \\ $$
Commented by Dwaipayan Shikari last updated on 23/Jun/20
Great question.Although  sir i am a student
$${Great}\:{question}.{Although}\:\:{sir}\:{i}\:{am}\:{a}\:{student} \\ $$

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