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Start by rationalizing each denominator. That is, for the denominator n + n + 1 , multiply the top and bottom of that fraction by n − n + 1 . In doing so, the given expression becomes
− 1 1 − 2 + − 1 2 − 3 + − 1 3 − 4 + ⋯ + − 1 8 − 9 ,
which simplifies to
( 2 − 1 ) + ( 3 − 2 ) + ( 4 − 3 ) + ⋯ + ( 9 − 8 ) = 9 − 1 = 2 .