I am reading an article and came across this expression and would appreciate some explanation.
"We have a function $$u(F)=2\Big[\frac{F-L}{L}-ln\frac{F}{L}\Big]$$
Since $u''(F)>0$ a Taylor series expansion with second-order remainder of $u(F_n)$ about $F_n=L$ implies $$u(F_n)=\int_0^L\frac{2}{K^2}(K-F_n)^+dK+\int_L^\infty\frac{2}{K^2}(F_n-K)^+dK$$"
Now, with what I know I would have the Taylor series to be $$\frac{2}{L^2}(L-F_n)$$ I do not see how the integrals play a part in the series expansion. If someone could shed some light over this it would be extremely helpful. Thanks!