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x + x^2 + x^4 + x^8 +... + x^{2^m}< 2010 , when x = 1 - 2^{-2009}

Source: Norwegian Mathematical Olympiad 2009 - Abel Competition p4a

September 5, 2019
algebrainequalitiesSum

Problem Statement

Let x=1−2−2009x = 1 - 2^{-2009}. Show that x+x2+x4+x8+...+x2m<2010x + x^2 + x^4 + x^8 +... + x^{2^m}< 2010 for all positive integers mm.
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