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Vojtěch Jarník IMC
2022 VJIMC
3
limit sum integral function
limit sum integral function
Source: VJIMC 2022 1.3
April 11, 2022
calculus
integration
function
Problem Statement
Let
f
:
[
0
,
1
]
→
R
f:[0,1]\to\mathbb R
f
:
[
0
,
1
]
→
R
be a given continuous function. Find the limit
lim
n
→
∞
(
n
+
1
)
∑
k
=
0
n
∫
0
1
x
k
(
1
−
x
)
n
−
k
f
(
x
)
d
x
.
\lim_{n\to\infty}(n+1)\sum_{k=0}^n\int^1_0x^k(1-x)^{n-k}f(x)dx.
n
→
∞
lim
(
n
+
1
)
k
=
0
∑
n
∫
0
1
x
k
(
1
−
x
)
n
−
k
f
(
x
)
d
x
.
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