## Estimation of Taylor Series

Suppose

Write

We call

the th degree Taylor polynomial. Notice that

if and only if

We would like to estimate with . We need an estimate for .

Theorem 6.7.1 (Taylor's theorem)   If for , then

for $|x-a|&le#leq;d$.

For example, if , this says that

i.e.,

which should look familiar from a previous class (Mean Value Theorem).

Applications:

1. One can use Theorem  to prove that functions converge to their Taylor series.

2. Returning to the relativity example above, we apply Taylor's theorem with and . With and any number such that , we have

For example, if we assume that we use

Using , we get

Thus for    mph, then the error in throwing away relativistic factors is . This is like 200 feet out of the distance to the sun (93 million miles). So relativistic and Newtonian kinetic energies are almost the same for reasonable speeds.

William Stein 2006-03-15