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Question Number 30436 by abdo imad last updated on 22/Feb/18

$$let\phi \left(x\right)=1-2^{1-x}provethat$$$$\phi \left(x\right)=(x-1)ln2-\frac{{\left(ln2\right)}^2}{2}{(x-1)}^2+o\left({(x-1)}^2\right).$$

Commented by prof Abdo imad last updated on 22/Feb/18

$$x\to 1$$

Answered by sma3l2996 last updated on 24/Feb/18

$$2^t=e^{tln\left(2\right)}=1+tln\left(2\right)+\frac{{\left(tln2\right)}^2}{2!}+o\left(t^2\right)$$$$so{2}^{1-x}=1+(1-x)ln\left(2\right)+\frac{{\left(ln2\right)}^2}{2}{(1-x)}^2+o\left({(1-x)}^2\right)$$$$1-2^{1-x}=-(1-x)ln\left(2\right)-\frac{{\left(ln2\right)}^2}{2}{(1-x)}^2+o\left({(x-1)}^2\right)$$$$\phi \left(x\right)=(x-1)ln\left(2\right)-\frac{{\left(ln2\right)}^2}{2}{(x-1)}^2+o\left({(x-1)}^2\right)$$

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