For small x, Taylor Series to determine constants n and C in the approximation

Question

cos(x) - e^(-(x^2)/2)

I have tried writing the taylor expansion for cosine and e but do not know how to combine them in one. In addition to this, i do not know how to approach the problem whilst determining n and C.

I appreciate any help.

Answer 1

Hint

Just as you did, start with the basic series $$\cos(x)=\sum_{n=0}^{\infty}\frac{(-1)^n}{(2n)!}x^{2n}$$ and $$e^{-y}=\sum_{n=0}^{\infty}\frac{(-1)^n}{n!}y^n$$ In the second one, replace $y$ by $\frac{x^2}{2}$ and now combine them for your expression.