MathDB
Problems
Contests
National and Regional Contests
Japan Contests
Today's Calculation Of Integral
2012 Today's Calculation Of Integral
783
783
Part of
2012 Today's Calculation Of Integral
Problems
(1)
Today's calculation of Integral 783
Source: 2012 Gakusyuin University entrance exam/Science
2/9/2012
Define a sequence
a
1
=
0
,
1
1
−
a
n
+
1
−
1
1
−
a
n
=
2
n
+
1
(
n
=
1
,
2
,
3
,
⋯
)
a_1=0,\ \frac{1}{1-a_{n+1}}-\frac{1}{1-a_n}=2n+1\ (n=1,\ 2,\ 3,\ \cdots)
a
1
=
0
,
1
−
a
n
+
1
1
−
1
−
a
n
1
=
2
n
+
1
(
n
=
1
,
2
,
3
,
⋯
)
.(1) Find
a
n
a_n
a
n
.(2) Let
b
k
=
k
+
1
k
(
1
−
a
k
+
1
)
{b_k=\sqrt{\frac{k+1}{k}}\ (1-\sqrt{a_{k+1}}})
b
k
=
k
k
+
1
(
1
−
a
k
+
1
)
for
k
=
1
,
2
,
3
,
⋯
k=1,\ 2,\ 3,\ \cdots
k
=
1
,
2
,
3
,
⋯
.Prove that
∑
k
=
1
n
b
k
<
2
−
1
\sum_{k=1}^n b_k<\sqrt{2}-1
∑
k
=
1
n
b
k
<
2
−
1
for each
n
n
n
.Last Edited
calculus
integration
calculus computations