The standard proof for Jensen's inequality using taylor expansion around a point $x_0$ involves using only first 3 terms of the Taylor series till $f^{\prime \prime}(x)$. Why are we able to ignore the higher degree terms in the proof? As far as I am aware, there is no assumption regarding the points being close so that the higher order terms vanish. The proof is listed here (page 24) - https://web.cse.msu.edu/~cse842/Papers/CoverThomas-Ch2.pdf
Any insight on this is appreciated.