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National and Regional Contests
Moldova Contests
Moldova National Olympiad
2006 Moldova National Olympiad
11.1
Trigonometric limit
Trigonometric limit
Source: Moldavian MO 2006
March 19, 2006
limit
trigonometry
algebra proposed
algebra
Problem Statement
Let
n
∈
N
∗
n\in\mathbb{N}^*
n
∈
N
∗
. Prove that
lim
x
→
0
(
1
+
x
2
)
n
+
1
−
∏
k
=
1
n
cos
k
x
x
∑
k
=
1
n
sin
k
x
=
2
n
2
+
n
+
12
6
n
.
\lim_{x\to 0}\frac{ \displaystyle (1+x^2)^{n+1}-\prod_{k=1}^n\cos kx}{ \displaystyle x\sum_{k=1}^n\sin kx}=\frac{2n^2+n+12}{6n}.
x
→
0
lim
x
k
=
1
∑
n
sin
k
x
(
1
+
x
2
)
n
+
1
−
k
=
1
∏
n
cos
k
x
=
6
n
2
n
2
+
n
+
12
.
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