Completely Convex Functions and Convergence

A function $f(x)$ is completely convex (c.c.) on $[0,1]$ if $( - 1)^k f^{(2k)} (x) \geqq 0$ for $k \geqq 0$ and all $x$ in $[0,1]$. This paper studies the convergence of the partial sums of the Maclaurin series of the function; in particular, how quickly the partial sums turn into a c.c. function. It is shown that no matter where the series is truncated, the resulting partial sum is a completely convex function in at least the interval $[0,{{\sqrt {10} } / 5}]$.