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2018 VTRMC
5
Bounding an Integral Sequence
Bounding an Integral Sequence
Source: 2018 VTRMC P5
January 8, 2023
calculus
integration
VTRMC
college contests
Sequences
Problem Statement
For
n
∈
N
n \in \mathbb{N}
n
∈
N
, let
a
n
=
∫
0
1
/
n
∣
1
+
e
i
t
+
e
2
i
t
+
⋯
+
e
n
i
t
∣
d
t
a_n = \int _0 ^{1/\sqrt{n}} | 1 + e^{it} + e^{2it} + \dots + e^{nit} | \ dt
a
n
=
∫
0
1/
n
∣1
+
e
i
t
+
e
2
i
t
+
⋯
+
e
ni
t
∣
d
t
. Determine whether the sequence
(
a
n
)
=
a
1
,
a
2
,
…
(a_n) = a_1, a_2, \dots
(
a
n
)
=
a
1
,
a
2
,
…
is bounded.
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