Question 2.17: Prove that ξ = 1 for hydrogen....

Let x = 1/α Then from Eq. (2.56) \xi=1+\frac{1}{1-x}\ln(x) Since for hydrogen \alpha\rightarrow0\, , x\to{\infty} Thus \xi=1+lim_{x\to \infty}{\frac{1}{1-x}}\ln(x) Use L’Hospital’s rule

lim_{x\to \infty}{\frac{1}{1-x}}\ln(x)= lim_{x\to \infty}{\frac{{\frac{d}{d x}}\ln(x)}{{\frac{d}{d x}}(1-x)}}= lim_{x\to \infty}{\frac{1/x}{-1}}=0  Thus  ξ = 1