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Jozsef Wildt International Math Competition
2019 Jozsef Wildt International Math Competition
W. 7
Find the value of this limit
Find the value of this limit
Source: 2019 Jozsef Wildt International Math Competition-W. 7
May 18, 2020
limit
integration
Summation
Sequences
Harmonic Numbers
Problem Statement
If
Ω
n
=
∑
k
=
1
n
(
∫
−
1
k
1
k
(
2
x
10
+
3
x
8
+
1
)
cos
−
1
(
k
x
)
d
x
)
\Omega_n=\sum \limits_{k=1}^n \left(\int \limits_{-\frac{1}{k}}^{\frac{1}{k}}(2x^{10} + 3x^8 + 1)\cos^{-1}(kx)dx\right)
Ω
n
=
k
=
1
∑
n
−
k
1
∫
k
1
(
2
x
10
+
3
x
8
+
1
)
cos
−
1
(
k
x
)
d
x
Then find
Ω
=
lim
n
→
∞
(
Ω
n
−
π
H
n
)
\Omega=\lim \limits_{n\to \infty}\left(\Omega_n-\pi H_n\right)
Ω
=
n
→
∞
lim
(
Ω
n
−
π
H
n
)
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