Answer
$\Sigma_{n=0}^\infty \dfrac{(-1)^n (x)^{10n/3}}{(2n)!}$
Work Step by Step
Since, we know that the Taylor Series for $\cos x$ is defined as:
$ \cos x=\Sigma_{n=0}^\infty \dfrac{(-1)^n x^{2n}}{(2n)!}=1-\dfrac{x^2}{2!}+\dfrac{x^4}{4!}-...$
Now, $ \cos (x^{5/3})=\Sigma_{n=0}^\infty \dfrac{(-1)^n (x^{5/3})^{2n}}{(2n)!}=1-\dfrac{(x^{5/3})^2}{2!}+\dfrac{(x^{5/3})^4}{4!}-...$
or, $=\Sigma_{n=0}^\infty \dfrac{(-1)^n (x)^{10n/3}}{(2n)!}$
