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Problems
Contests
National and Regional Contests
PEN Problems
PEN G Problems
23
23
Part of
PEN G Problems
Problems
(1)
G 23
Source:
5/25/2007
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.
calculus
derivative
Irrational numbers