The Repeated Power Function

n

100

x

0

0

dx

1.5

y

5

Leonhard Euler proved that the function

h

n

(x)=

x

.

x

, where

x

appears

n

times, approaches a limit

h(x)

for

-e

e

<x<

1/e

e

as

n∞

. (The red points indicate the limits at the endpoints of the interval of convergence.) This Demonstration shows the values of

h

n

(x)

for

n=1

to

1000

. You can zoom in to see the limits at

x=

-

e

and

x=

1/e

e

. The expression bifurcates at the lower endpoint (

x=

-

e

) with

h

n

(

-

e

)1/e

. The sequence

h

2n

(x)

converges to 1 as

x0

from above while

h

2n+1

(x)

converges to 0. Beyond the upper endpoint (

x>

1/e

e

) the expression rapidly grows to infinity.