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National and Regional Contests
PEN Problems
PEN F Problems
14
14
Part of
PEN F Problems
Problems
(1)
F 14
Source:
5/25/2007
Let
k
k
k
and
m
m
m
be positive integers. Show that
S
(
m
,
k
)
=
∑
n
=
1
∞
1
n
(
m
n
+
k
)
S(m, k)=\sum_{n=1}^{\infty}\frac{1}{n(mn+k)}
S
(
m
,
k
)
=
n
=
1
∑
∞
n
(
mn
+
k
)
1
is rational if and only if
m
m
m
divides
k
k
k
.
rational numbers