Find the power series for $f(x) = \frac{\cos(x^3)}{2x^2}$

Question

I'm pretty sure if it were just $\cos(x^3)$ i could subsititue $x^3$ for $x$, everywhere in the known series, but what do I do because it's divided by $2x^2$?

Answer 1

We know that $$\cos x=\sum^\infty_{k=0}\dfrac{(-1)^kx^{2k}}{(2k)!}$$ Substitute $x\to x^3$, and divide by $2x^2$ to get (Note that the term $2x^2$ has no dependence on $k$) $$\dfrac{\cos(x^3)}{2x^2}=\sum^\infty_{k=0}\dfrac{(-1)^kx^{6k-2}}{2(2k)!}$$

Answer 2

You were close. Just divide the resulting sequence by $2x^2$ to get $$ \frac1{2x^2}-\frac{x^6}{2\cdot 2 x^2}+\dots $$