# How does one find the Taylor Series for $sin(x)$ at the 3rd Taylor Polynomial at $x = \frac{2}{3}$ with an upper bond?

Write down the Taylor Series for f(x) = sin(x), and find the value of the 3rd Taylor Polynomial P_3 at x = 2/3. Then find a reasonable upper bound for the error using P_3(2/3) for the value of sin(2/3)

Thank you

• What Lagrange's theorem tells you about the difference between $\sin(x)$ and $P_3(x)$? – Jack D'Aurizio May 2 '17 at 17:38
• Or, equivalently, do you see a simple way for providing an upper bound for $$\left|\sum_{n\geq 2}\frac{(2/3)^{2n+1}(-1)^n}{(2n+1)!}\right|$$ ? – Jack D'Aurizio May 2 '17 at 17:39
• Bounds on alternating series? – Simply Beautiful Art May 2 '17 at 18:00