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Today's Calculation Of Integral
2005 Today's Calculation Of Integral
76
Today's calculation of Integral 76
Today's calculation of Integral 76
Source: 1992 Toyama University
August 26, 2005
calculus
integration
function
limit
inequalities
calculus computations
Problem Statement
The function
f
n
(
x
)
(
n
=
1
,
2
,
⋯
)
f_n (x)\ (n=1,2,\cdots)
f
n
(
x
)
(
n
=
1
,
2
,
⋯
)
is defined as follows.
f
1
(
x
)
=
x
,
f
n
+
1
(
x
)
=
2
x
n
+
1
−
x
n
+
1
2
∫
0
1
f
n
(
t
)
d
t
(
n
=
1
,
2
,
⋯
)
f_1 (x)=x,\ f_{n+1}(x)=2x^{n+1}-x^n+\frac{1}{2}\int_0^1 f_n(t)\ dt\ \ (n=1,2,\cdots)
f
1
(
x
)
=
x
,
f
n
+
1
(
x
)
=
2
x
n
+
1
−
x
n
+
2
1
∫
0
1
f
n
(
t
)
d
t
(
n
=
1
,
2
,
⋯
)
Evaluate
lim
n
→
∞
f
n
(
1
+
1
2
n
)
\lim_{n\to\infty} f_n \left(1+\frac{1}{2n}\right)
n
→
∞
lim
f
n
(
1
+
2
n
1
)
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