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 the$x_{1}$ 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