Template:Sequence of the Day for May 22

Intended for: May 22, 2013

Timetable

• First draft entered by Alonso del Arte on March 23, 2012 ✓
• Draft reviewed by Daniel Forgues on May 13, 2012; May 21, 2015 ✓
• Draft to be approved by April 22, 2013

Yesterday's SOTD * Tomorrow's SOTD

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A100943: Decimal expansion of the infinitely nested radical $\textstyle {\sqrt {e+{\sqrt {e+{\sqrt {e+\cdots }}}}}}$, where

e

is Euler’s number.

2.222870229721...

This number is the positive solution from

y = 1 ± 2√  1 + 4 e 2

to the equation

y 2  −  y  −  e = 0

, the other solution being

1  −  y =  − 1.222870229721...

, which has the same fractional part (so to speak, since those numbers are transcendental). By looking at the infinitely nested surd, it is trivial to see that

y 2  −  y = e

. One can easily verify that

(1  −  y) 2  −  (1  −  y) = (1  −  2 y  +  y 2 )  −  (1  −  y) = y 2  −  y = e

.