((1/2))^(((1/3))^((1/4)....∞) ) =?


Question and Answers Forum

All Questions Topic List
Others Questions
Previous in All Question Next in All Question
Previous in Others Next in Others
Question Number 99975 by Dwaipayan Shikari last updated on 24/Jun/20

${(\frac{1}{2})}^{{(\frac{1}{3})}^{\frac{1}{4} . . . . \infty}} = ?$
	Answered by bachamohamed last updated on 24/Jun/20

	$s o l u t i o n :$ $o n a a_{n} = {(\frac{1}{2})}^{{(\frac{1}{3})}^{{(\frac{1}{4})}^{. . . . . . . . (\frac{1}{n + 1})}}} . a_{1} = (\frac{1}{2}) . a_{2} = {(\frac{1}{2})}^{(\frac{1}{3})} = e^{\frac{1}{3} l n \frac{1}{2}}$ $a_{3} = {(\frac{1}{2})}^{{(\frac{1}{3})}^{(\frac{1}{4})}} = e^{\frac{1}{4} l n ({(\frac{1}{2})}^{(\frac{1}{3})})} = e^{\frac{1}{4} l n (a_{2})} = e^{\frac{1}{4} l n (e^{\frac{1}{3} l n (\frac{1}{2})})} = e^{\frac{2}{4!} l n (\frac{1}{2})}$ $a_{4} = {(\frac{1}{2})}^{{(\frac{1}{3})}^{{(\frac{1}{4})}^{(\frac{1}{5})}}} = e^{\frac{1}{5} l n ({(\frac{1}{2})}^{{(\frac{1}{3})}^{(\frac{1}{4})}})} = e^{\frac{1}{5} l n (a_{3})} = e^{\frac{1}{5} l n (e^{\frac{2}{4!} l n (\frac{1}{2})})} = e^{\frac{2}{5!} l n (\frac{1}{2})}$ $a_{5} = {(\frac{1}{2})}^{{(\frac{1}{3})}^{{(\frac{1}{4})}^{{(\frac{1}{5})}^{(\frac{1}{6})}}}} = e^{\frac{1}{6} l n (a_{4})} = e^{\frac{1}{6} l n (e^{\frac{2}{5!} l n (\frac{1}{2})})} = e^{\frac{2}{6!} l n (\frac{1}{2})}$ $. . . .$ $. . .$ $a_{n} = {(\frac{1}{2})}^{{(\frac{1}{3})}^{. . . . . (\frac{1}{n + 1})}} = e^{\frac{1}{(n + 1)} l n (a_{n - 1})} = e^{\frac{1}{(n + 1)} l n (e^{\frac{2}{n!} l n (\frac{1}{2})})} = e^{\frac{2}{(n + 1)!} l n (\frac{1}{2})}$ $e t : n > 0 (n + 1) < (n + 1)! < {(n + 1)}^{(n + 1)} \Rightarrow n > 0 \frac{2}{(n + 1)} > \frac{2}{(n + 1)!} > \frac{2}{{(n + 1)}^{(n + 1)}}$ $n > 0 {(\frac{1}{2})}^{(\frac{2}{n + 1})} < {(\frac{1}{2})}^{(\frac{2}{(n + 1)!})} < {(\frac{1}{2})}^{\frac{2}{{(n + 1)}^{(n + 1)}}} \Rightarrow n > 0 . {(\frac{1}{2})}^{(\frac{2}{n + 1})} < a_{n} = e^{\frac{2}{(n + 1)!} l n (\frac{1}{2})} < {(\frac{1}{2})}^{\frac{2}{{(n + 1)}^{(n + 1)}}}$ $Double subscripts: use braces to clarify$ $Double subscripts: use braces to clarify$
	Commented by Dwaipayan Shikari last updated on 24/Jun/20

	$I h a d a t h o u g h t l i k e y o u . O u r t h i n k i n g i s m a t c h e d . G r e a t s i r$
	Commented by Dwaipayan Shikari last updated on 24/Jun/20

	$Just matched!! But i was not so sure about answer . Sir you create my belief$
	Commented by bachamohamed last updated on 24/Jun/20

	${t h a n k}^{'} s$
Terms of Service
Privacy Policy
Contact: info@tinkutara.com

Question and Answers Forum