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SEEMOUS
2019 SEEMOUS
4
SEEMOUS 2019, problem 4
SEEMOUS 2019, problem 4
Source:
March 18, 2019
real analysis
college contests
Problem Statement
(a) Let
n
n
n
is a positive integer. Calculate
∫
0
1
x
n
−
1
ln
x
d
x
\displaystyle \int_0^1 x^{n-1}\ln x\,dx
∫
0
1
x
n
−
1
ln
x
d
x
.\\(b) Calculate
∑
n
=
0
∞
(
−
1
)
n
(
1
(
n
+
1
)
2
−
1
(
n
+
2
)
2
+
1
(
n
+
3
)
2
−
…
)
.
\displaystyle \sum_{n=0}^{\infty}(-1)^n\left(\frac{1}{(n+1)^2}-\frac{1}{(n+2)^2}+\frac{1}{(n+3)^2}-\dots \right).
n
=
0
∑
∞
(
−
1
)
n
(
(
n
+
1
)
2
1
−
(
n
+
2
)
2
1
+
(
n
+
3
)
2
1
−
…
)
.
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