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Verified answer
(1 - 1/(10)²)(1 - 1/(11)²)...(1 - 1/(100)²= ((10)²/(10)² - 1/(10)²) ((11)²/(11)² - 1/(11)²)...((100)²/(100)² - 1/(100)²)
= ((10²-1)/(10)²) ((11²-1)/(11)²) ... ((100²-1)/(100)²)
= ((10+1)(10-1)/10²)((11+1)(11-1)/11²)...((100+1)(100-1)/100²)
= ((11)(9)/(10)(10)) ((12)(10)/(11)(11))...((101)(99)/(100)(100))
= (9/10)(101/100)
= 909/1000
E