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PEN G Problems
27
G 27
G 27
Source:
May 25, 2007
limit
calculus
function
derivative
integration
Irrational numbers
Problem Statement
Let
1
<
a
1
<
a
2
<
⋯
1<a_{1}<a_{2}<\cdots
1
<
a
1
<
a
2
<
⋯
be a sequence of positive integers. Show that
2
a
1
a
1
!
+
2
a
2
a
2
!
+
2
a
3
a
3
!
+
⋯
\frac{2^{a_{1}}}{{a_{1}}!}+\frac{2^{a_{2}}}{{a_{2}}!}+\frac{2^{a_{3}}}{{a_{3}}!}+\cdots
a
1
!
2
a
1
+
a
2
!
2
a
2
+
a
3
!
2
a
3
+
⋯
is irrational.
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