91Ó°ÊÓ

Question: Show that the remainder of the Taylor series for the function \(f(x) = \sin{x}\) around the point \(a = 0\) converges to 0 for all x in its interval of convergence. Answer: We can show that the limit of the remainder converges to 0 for all x by using the Lagrange form of the remainder: $$\lim_{n \rightarrow \infty} R_n(x) = \lim_{n \rightarrow \infty} \frac{f^{(n+1)}(c)}{(n+1)!} x^{n+1}$$ Since the maximum value of \(|f^{(n+1)}(c)|\) is 1, we can apply the limit and get: $$\lim_{n \rightarrow \infty} \left|R_n(x) \right| \le \lim_{n \rightarrow \infty} \frac{|x|^{n+1}}{(n+1)!} = 0$$ As the limit is less than or equal to 0 for all \(x\) in the interval of convergence, we can conclude that the limit is indeed 0: $$\lim_{n \rightarrow \infty} R_n(x) = 0$$