MathDB
Problems
Contests
National and Regional Contests
Japan Contests
Today's Calculation Of Integral
2005 Today's Calculation Of Integral
30
30
Part of
2005 Today's Calculation Of Integral
Problems
(1)
Today's calculation of Integral 30
Source: 1990 Shibaura Institute of Technology
5/31/2005
A sequence
{
a
n
}
\{a_n\}
{
a
n
}
is defined by
a
n
=
∫
0
1
x
3
(
1
−
x
)
n
d
x
(
n
=
1
,
2
,
3.
⋯
)
a_n=\int_0^1 x^3(1-x)^n dx\ (n=1,2,3.\cdots)
a
n
=
∫
0
1
x
3
(
1
−
x
)
n
d
x
(
n
=
1
,
2
,
3.
⋯
)
Find the constant number
c
c
c
such that
∑
n
=
1
∞
(
n
+
c
)
(
a
n
−
a
n
+
1
)
=
1
3
\sum_{n=1}^{\infty} (n+c)(a_n-a_{n+1})=\frac{1}{3}
∑
n
=
1
∞
(
n
+
c
)
(
a
n
−
a
n
+
1
)
=
3
1
calculus
integration
algebra
partial fractions
calculus computations