Trouble with simple elasticity derivation

[K29]

Deriving the basics of infinitesmal elasticity, I have a proof regarding the physical significance of the strain tensor E_{11}

So we are considering a material line element parallel with the x_{1} axis being deformed.

At a point in the proof we have

du_{1} = u_{1}(x_{1}+dL_{0}, x_{2}, x_{3})-u_{1}(x_{1}, x_{2}, x_{3})

This is just saying we have a deformation happening along thex_{1} axis.

where u is a displacement vector: u=R-r where r is the intiial position of an element in the elastic material and R is the same element's position while deformed. u_{1} is parallel to the x_{1} axis

L_{0} is a material line element parallel to the x_{1} axis. So considering that line element: dr = dL_{0}\hat{i}_{1}

Anyway they have the following step:

du_{1} = u_{1}(x_{1}+dL_{0}, x_{2}, x_{3})-u_{1}(x_{1}, x_{2}, x_{3})

=u_{1}(x_{1},x_{2}, x_{3})+\frac{\partial u_{1}}{\partial x_{1}}dL_{0}+O(dL_{0})^{2}-u_{1}(x_{1}, x_{2}, x_{3})

I am stuck with what happened there. Any help with what happened there would be appreciated.

 

[K29]

I see on wikipedia that displacement functions can be represented as a sum of an infinite series. It does not give further details. Is it perhaps this? how so? Please help