MathDB
Problems
Contests
Undergraduate contests
SEEMOUS
2020 SEEMOUS
Problem 2
SEEMOUS 2020 P2
SEEMOUS 2020 P2
Source:
May 2, 2020
calculus
Problem Statement
Let
k
>
1
k>1
k
>
1
be a real number. Calculate: (a)
L
=
lim
n
→
∞
∫
0
1
(
k
x
n
+
k
−
1
)
n
d
x
.
L=\lim_{n\to \infty} \int_0^1\left( \frac{k}{\sqrt[n]{x}+k-1}\right)^n\text{d} x.
L
=
lim
n
→
∞
∫
0
1
(
n
x
+
k
−
1
k
)
n
d
x
.
(b)
lim
n
→
∞
n
[
L
−
∫
0
1
(
k
x
n
+
k
−
1
)
n
d
x
]
.
\lim_{n\to \infty} n\left\lbrack L- \int_0^1\left( \frac{k}{\sqrt[n]{x}+k-1}\right)^n\text{d} x\right\rbrack.
lim
n
→
∞
n
[
L
−
∫
0
1
(
n
x
+
k
−
1
k
)
n
d
x
]
.
Back to Problems
View on AoPS