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Today's Calculation Of Integral
2005 Today's Calculation Of Integral
36
Today's calculation of Integral 36
Today's calculation of Integral 36
Source: 2004 Nagoya University
June 6, 2005
calculus
integration
algebra
polynomial
limit
calculus computations
Problem Statement
A sequence of polynomial
f
n
(
x
)
(
n
=
0
,
1
,
2
,
⋯
)
f_n(x)\ (n=0,1,2,\cdots)
f
n
(
x
)
(
n
=
0
,
1
,
2
,
⋯
)
satisfies
f
0
(
x
)
=
2
,
f
1
(
x
)
=
x
f_0(x)=2,f_1(x)=x
f
0
(
x
)
=
2
,
f
1
(
x
)
=
x
,
f
n
(
x
)
=
x
f
n
−
1
(
x
)
−
f
n
−
2
(
x
)
,
(
n
=
2
,
3
,
4
,
⋯
)
f_n(x)=xf_{n-1}(x)-f_{n-2}(x),\ (n=2,3,4,\cdots)
f
n
(
x
)
=
x
f
n
−
1
(
x
)
−
f
n
−
2
(
x
)
,
(
n
=
2
,
3
,
4
,
⋯
)
Let
x
n
(
n
≧
2
)
x_n\ (n\geqq 2)
x
n
(
n
≧
2
)
be the maximum real root of the equation
f
n
(
x
)
=
0
(
∣
x
∣
≦
2
)
f_n(x)=0\ (|x|\leqq 2)
f
n
(
x
)
=
0
(
∣
x
∣
≦
2
)
Evaluate
lim
n
→
∞
n
2
∫
x
n
2
f
n
(
x
)
d
x
\lim_{n\to\infty} n^2 \int_{x_n}^2 f_n(x)dx
n
→
∞
lim
n
2
∫
x
n
2
f
n
(
x
)
d
x
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