Approximation of $\sin(\pi)$ by $f(x)=\sin(3x)$
Solution 1:
Evaluate your Taylor polynomials at $x=\pi$: \begin{align} T_n(\pi) &= \sum_{k=0}^n \frac{(-1)^{k+1}3^{2k+1}}{(2k+1)!} \biggl( \pi - \frac{\pi}{3} \biggr)^{2k+1} \\ &= \sum_{k=0}^n \frac{(-1)^{k+1}2^{2k+1}\pi^{2k+1}}{(2k+1)!} \\ \end{align} Of course, you know that $\sin \pi = 0$, so these partial sums converge to $0$, but it's interesting to see how they converge.