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Vojtěch Jarník IMC
1997 VJIMC
Problem 4-M
infinite factorial sum with trig expression
infinite factorial sum with trig expression
Source: VJIMC 1997 2.4-M
October 9, 2021
trigonometry
Summation
real analysis
limits
factorial
Problem Statement
Prove that
∑
n
=
1
∞
n
2
(
7
n
)
!
=
1
7
3
∑
k
=
1
2
∑
j
=
0
6
e
cos
(
2
π
j
/
7
)
⋅
cos
(
2
k
π
j
7
+
sin
2
π
j
7
)
.
\sum_{n=1}^\infty\frac{n^2}{(7n)!}=\frac1{7^3}\sum_{k=1}^2\sum_{j=0}^6e^{\cos(2\pi j/7)}\cdot\cos\left(\frac{2k\pi j}7+\sin\frac{2\pi j}7\right).
n
=
1
∑
∞
(
7
n
)!
n
2
=
7
3
1
k
=
1
∑
2
j
=
0
∑
6
e
c
o
s
(
2
πj
/7
)
⋅
cos
(
7
2
kπj
+
sin
7
2
πj
)
.
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