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SEEMOUS
2012 SEEMOUS
Problem 4
two integrals, limits of n to infinity
two integrals, limits of n to infinity
Source: SEEMOUS 2012 P4
June 9, 2021
calculus
integration
Problem Statement
a) Compute
lim
n
→
∞
n
∫
0
1
(
1
−
x
1
+
x
)
n
d
x
.
\lim_{n\to\infty}n\int^1_0\left(\frac{1-x}{1+x}\right)^ndx.
n
→
∞
lim
n
∫
0
1
(
1
+
x
1
−
x
)
n
d
x
.
b) Let
k
≥
1
k\ge1
k
≥
1
be an integer. Compute
lim
n
→
∞
n
k
+
1
∫
0
1
(
1
−
x
1
+
x
)
n
x
k
d
x
.
\lim_{n\to\infty}n^{k+1}\int^1_0\left(\frac{1-x}{1+x}\right)^nx^kdx.
n
→
∞
lim
n
k
+
1
∫
0
1
(
1
+
x
1
−
x
)
n
x
k
d
x
.
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