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SEEMOUS
2021 SEEMOUS
Problem 4
Convergence of a strange sequence
Convergence of a strange sequence
Source: 2021 SEEMOUS, P4
July 23, 2021
Convergence
real analysis
Problem Statement
For
p
∈
R
p \in \mathbb{R}
p
∈
R
, let
(
a
n
)
n
≥
1
(a_n)_{n \ge 1}
(
a
n
)
n
≥
1
be the sequence defined by
a
n
=
1
n
p
∫
0
n
∣
sin
(
π
x
)
∣
x
d
x
.
a_n=\frac{1}{n^p} \int_0^n |\sin( \pi x)|^x \mathrm dx.
a
n
=
n
p
1
∫
0
n
∣
sin
(
π
x
)
∣
x
d
x
.
Determine all possible values of
p
p
p
for which the series
∑
n
=
1
∞
a
n
\sum_{n=1}^\infty a_n
∑
n
=
1
∞
a
n
converges.
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