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Question Number 30436 by abdo imad last updated on 22/Feb/18
let ϕ(x)=1−2^(1−x) prove that ϕ(x)=(x−1)ln2 −(((ln2)^2 )/2)(x−1)^2 +o((x−1)^2 ).
letφ(x)=1−21−xprovethatφ(x)=(x−1)ln2−(ln2)22(x−1)2+o((x−1)2).
Commented by prof Abdo imad last updated on 22/Feb/18
x→1
x→1
Answered by sma3l2996 last updated on 24/Feb/18
2^t =e^(tln(2)) =1+tln(2)+(((tln2)^2 )/(2!))+o(t^2 ) so 2^(1−x) =1+(1−x)ln(2)+(((ln2)^2 )/2)(1−x)^2 +o((1−x)^2 ) 1−2^(1−x) =−(1−x)ln(2)−(((ln2)^2 )/2)(1−x)^2 +o((x−1)^2 ) ϕ(x)=(x−1)ln(2)−(((ln2)^2 )/2)(x−1)^2 +o((x−1)^2 )
2t=etln(2)=1+tln(2)+(tln2)22!+o(t2)so21−x=1+(1−x)ln(2)+(ln2)22(1−x)2+o((1−x)2)1−21−x=−(1−x)ln(2)−(ln2)22(1−x)2+o((x−1)2)φ(x)=(x−1)ln(2)−(ln2)22(x−1)2+o((x−1)2)

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