Review : Taylor Series — Question 9

Question 9 – Solution

Strategy. Match coefficients, then examine the first coefficient on which the competing approximations disagree.

Step 1: Solve the matching equations. Near $0$, $Q=1+(a-b)x^2+(b^2-ab)x^4+\cdots$. Thus $a-b=-1/2$ and $-b(a-b)=1/24$, giving $$\boxed{b=\frac 1{12},\quad a=-\frac 5{12},\quad Q(x)=\frac{12-5x^2}{12+x^2}.}$$ Since $12+x^2>0$, there are no real poles.

Step 2: Compare the leading errors. The rational geometric expansion has $x^6$ coefficient $-1/288$. Hence $$\cos x-T_4(x)=-\frac{x^6}{720}+O(x^8),\qquad \cos x-Q(x)=\frac{x^6}{480}+O(x^8).$$ Therefore $\boxed{\lim_{x\to 0}\frac{|\cos x-Q(x)|}{|\cos x-T_4(x)|}=3/2}$. In particular, the rational approximation is worse for all sufficiently small nonzero $x$.

Step 3: Make an exact comparison at one. Alternating bounds give $$T_6(1)=\frac{389}{720}<\cos 1<T_4(1)=\frac{13}{24},\qquad Q(1)=\frac 7{13}.$$ Here $Q(1)<389/720$, and the midpoint of the two approximations is $\tfrac 12(7/13+13/24)=337/624<389/720$. Thus $\cos 1$ lies above their midpoint and between them: $T_4(1)$ is strictly closer than $Q(1)$.

Step 4: State the proper conclusion. The claim fails both locally and at $x=1$. Matching four derivatives fixes contact at the center, not the next error coefficient or behavior far away. For $0<|x|\le 1$, alternating bounds give $\cos x-T_4(x)<0$ and $$\cos x-Q(x)\ge x^6\left(\frac 1{312}-\frac 1{720}\right)>0,$$ since $T_4-Q=x^6/[288(1+x^2/12)]$. These signs explain the plotted errors.