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okay weve got a big an interesting goal for this video and that is to classify all the real numbers between zero and one where this dialogue rhythm function in other words the sum as n goes from 1 to infinity of X to the N over N squared is nice and what I mean by nice is youve got some closed form in terms of elementary functions and so were going to use the following tools so were what I dont have written on the board is were gonna have an integral formula for this dialogue rhythm in other words for this sum so were gonna drive that first and then also were going to set this function L of x equal to this li-2 of X this dialogue rhythm function plus 1/2 natural log of x times natural log of 1 minus X thats going to be an important function to work with itll simplify some of the calculations and then were gonna prove this following lemma which has two parts and that is L of X plus L of 1 minus X is PI squared over 6 and also L of x equals L of x over 1 plus X plus 1/2 L of