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The Chinese Mathematics Competition
2021 The Chinese Mathematics Competition
Problem 10
Problem 10
Part of
2021 The Chinese Mathematics Competition
Problems
(1)
2021 CMC (Non-Math Major) Problem 10
Source: 2021 CMC (Non-Math Major) Problem 10
11/25/2022
Let
a
n
{a_n}
a
n
and
b
n
{b_n}
b
n
be positive real sequence that satisfy the following condition: (i)
a
1
=
b
1
=
1
a_1=b_1=1
a
1
=
b
1
=
1
(ii)
b
n
=
a
n
b
n
−
1
−
2
b_n=a_n b_{n-1}-2
b
n
=
a
n
b
n
−
1
−
2
(iii)
n
n
n
is an integer larger than
1
1
1
. Let
b
n
{b_n}
b
n
be a bounded sequence. Prove that
∑
n
=
1
∞
1
a
1
a
2
⋯
a
n
\sum_{n=1}^{\infty} \frac{1}{a_1a_2\cdots a_n}
∑
n
=
1
∞
a
1
a
2
⋯
a
n
1
converges. Find the value of the sum.
calculus