# Taylor Theorum

• Feb 26th 2006, 03:15 PM
me > jallison + brownie
Taylor Theorum
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
• Feb 26th 2006, 09:53 PM
CaptainBlack
Quote:

Originally Posted by me > jallison + brownie
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 an
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
$|x|<1$):

$
\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:

$
\left| \frac{x^5}{5} \right|=\frac{(0.4)^5}{5}\approx 0.002
$

RonL