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PEN G Problems
23
G 23
G 23
Source:
May 25, 2007
calculus
derivative
Irrational numbers
Problem Statement
Let
f
(
x
)
=
∏
n
=
1
∞
(
1
+
x
2
n
)
f(x)=\prod_{n=1}^{\infty} \left( 1 + \frac{x}{2^n} \right)
f
(
x
)
=
∏
n
=
1
∞
(
1
+
2
n
x
)
. Show that at the point
x
=
1
x=1
x
=
1
,
f
(
x
)
f(x)
f
(
x
)
and all its derivatives are irrational.
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