91Ó°ÊÓ

Question: Prove that the limit of the remainder term \(R_n(x)\) in the Taylor series centered at \(a = \pi/2\) for the function \(f(x) = \cos x\) is zero as \(n\) approaches infinity for all \(x\) in the interval of convergence. Answer: To prove this, we first determined the Taylor series for \(f(x)\) centered at \(a = \pi/2\), which is given by: $$T_{n}(x) = \sum_{k=0, k\, odd}^{n} \frac{(-1)^{(k-1)/2}(x-\pi/2)^k}{k!}$$ Next, we calculated the remainder term, \(R_n(x)\), using Taylor's theorem, and found that its absolute value can be bounded by: $$\left|R_n(x)\right|\leq \frac{(x-\pi/2)^{n+1}}{(n+1)!}$$ Finally, we showed that the limit of the absolute value of the remainder as \(n\) approaches infinity is zero since the factorial grows much faster than the power function, causing the fraction to tend to zero: $$\lim_{n \to \infty} \left|R_n(x)\right| \leq \lim_{n \to \infty} \frac{(x-\pi/2)^{n+1}}{(n+1)!} = 0$$ Thus, the limit of the remainder term, \(R_n(x)\), as \(n\) approaches infinity is indeed zero for all \(x\) in the interval of convergence.