In a Taylor polynomial, how would I find the upper bound for the error of the approximation: arctan(0.4) = 0.4 - (0.4)^3/3! . Thanks
Look at the form of the remainder of the truncated Taylor series. This is anOriginally Posted by me > jallison + brownie
alternating series (the sign of consecutive terms is alternately + and -) of
decreasing terms so the absolute error is less than the absolute value first
ignored term. In this case the Taylor series is (as we are interested in
$\displaystyle |x|<1$):
$\displaystyle
\arctan(x)=x-\frac{x^3}{3}+\frac{x^5}{5}-...
$
(note you have an error in your Taylor polynomial).
So in this case if we truncate after the second term the absolute error in the
approximation is less than:
$\displaystyle
\left| \frac{x^5}{5} \right|=\frac{(0.4)^5}{5}\approx 0.002
$
RonL