Sequence

[o.m.]

\( x(0)>0 \)

\( x(n+1)=x(n) - e^{\frac{-1}{x(n)^2}} \)

Prove that \( x(n)=\frac{1}{\sqrt{\ln n}} - \frac{3}{4}\frac{\ln(\ln n)}{(\ln n)^{3/2}} + \frac{u(n)\ln(\ln n)}{(\ln n)^{3/2}} \)

with u(n) tend to 0 when n tend to infinity.