The value of the infinite product below can be expressed as for coprime positive integers and Find .
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Let A k = ( k 2 − . 8 1 ) ( k 2 − 1 . 2 1 ) We are looking for P = k = 2 ∏ ∞ A k We will try and convert this into a telescoping product
A k = ( k + 0 . 9 ) ( k − 0 . 9 ) ( k + 1 . 1 ) ( k − 1 . 1 ) Now let
f ( k ) = k − 0 . 9 g ( k ) = k + 0 . 9 ........ the product transforms as P = k = 2 ∏ ∞ f ( k ) f ( k + 2 ) × g ( k ) g ( k − 2 ) P = ( f ( 2 ) . f ( 3 ) . f ( 4 ) . . . . f ( 4 ) . f ( 5 ) . f ( 6 ) . . . . . ) × ( g ( 2 ) . g ( 3 ) . g ( 4 ) . . . . . g ( 0 ) . g ( 1 ) . g ( 2 ) . . . . . )
All that remains is P = f ( 2 ) . f ( 3 ) g ( 0 ) . g ( 1 ) P = 1 . 1 × 2 . 1 0 . 9 × 1 . 9 P = 7 7 5 7
answer is 1 3 4