Similarly, what is the maximum value of this nested radical? I want to share a similar nested radical, but this time with varying breaking forces.
What is the minimum value of $$ f_ infty = frac {x} { sqrt {x sqrt[3]{x sqrt[4]{x cdots}}}} $$ Where do the radicals rise by one each time?
 Here is a plot of $ f_ {19} $, We can see that as $ x to 1 ^ + $, $ min f_ {19} to $ 1,7186 which is strange because the denominator can only accept the binary values $ 0 $ or $ 1 $ at the $ x = 1 $, The curve increases from monotonous $ 1 $ continue what is expected as the counter dominates.

Actually a simulation in PARI / GP up $ f_ {100} $ gives a minimum value of around $ 1,718,565That's a bit close $ e1 $although I strongly doubt it will ever reach that value.

Note that $ f_k $ is defined in $ (1, infty) $ for all positive integers $ k $but the curve is swinging wildly $ ( infty, 1) $,
Another interesting question: Why is the minimum value of $ f_k $ for big $ k $ not equal to the expected $ 0.1 $ or $ pm infty $? Can you manipulate? $ f_ infty $ so L & # 39; hopital can be used to determine the value of $ 1,718 cdots $?