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We consider the class of all analytic and locally univalent functions $f$ ofthe form $f(z)=z+\sum_{n=2}^\infty a_{2n-1} z^{2n-1}$, $|z|<1$, satisfying thecondition $$ {\rm Re}\,\left(1+\frac{zf^{\prime\prime}(z)}{f^\prime(z)}\right)>-\frac{1}{2}. $$ We show that every section$s_{2n-1}(z)=z+\sum_{k=2}^na_{2k-1}z^{2k-1}$, of $f$, is convex in the disk$|z|<\sqrt{2}/3$. We also prove that the radius $\sqrt{2}/3$ is best possible,i.e. the number $\sqrt{2}/3$ cannot be replaced by a larger one.

Radius of convexity of partial sums of odd functions in the close-to-convex family