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2021 CMC (Non-Math Major) Problem 10

Source: 2021 CMC (Non-Math Major) Problem 10

November 25, 2022

calculus

Problem Statement

Let

{a_n}

and

{b_n}

be positive real sequence that satisfy the following condition: (i)

a_1=b_1=1

(ii)

b_n=a_n b_{n-1}-2

(iii)

n

is an integer larger than

1

. Let

{b_n}

be a bounded sequence. Prove that

\sum_{n=1}^{\infty} \frac{1}{a_1a_2\cdots a_n}

converges. Find the value of the sum.