solve for x>0 (1+(1/x))^x >(5/2)


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Question Number 182644 by mr W last updated on 12/Dec/22

$s o l v e f o r x > 0$ ${(1 + \frac{1}{x})}^{x} > \frac{5}{2}$
Commented byFrix last updated on 12/Dec/22

$Obviously x > 1$
Commented bymr W last updated on 12/Dec/22

$i t s h o u l d n o t b e a t o o s p e c i a l c a s e,$ $t h e r e f o r e i h a v e m o d i f i e d t h e$ $q u e s t i o n .$
Commented byFrix last updated on 12/Dec/22

$x = \frac{1}{t}$ $t + 1 > {(\frac{5}{2})}^{t}$ $This should be possible with LambertW$
Commented byMJS_new last updated on 12/Dec/22
$I get t+1=q^t ⇒ t=−(1+((W(−((ln q)/q)))/(ln q))) q=(5/2) ⇔ { ((t=0)),((t≈.188114820)) :} ⇒ x≈5.31590237$
$I get$ $t + 1 = q^{t} \Rightarrow t = - (1 + \frac{W (- \frac{\ln q}{q})}{\ln q})$ $q = \frac{5}{2} \Leftrightarrow {\begin{cases} t = 0 \\ t \approx . 188114820 \end{cases} \Rightarrow x \approx 5.31590237$
Commented bymr W last updated on 13/Dec/22

$t h a n k s s i r s!$
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