Page 104 - Calculus for the Clueless, Calc II

4
4
Let's estimate. 0 < w < .1, so 1 < w + 1 < 1.1. And sooooo, 1/1.1 < 1/(w + 1) < 1; | R 3(.1) | = (.1) /4(W + 1) <
4
.1 /4 = .000025. Not bad!!
Example 40—

Let's do the same for f(x) = sin x except let a = 30º = π/6 with a polynomial of degree 3, and x = 32º. Hold on to
your hats, 'cause this is pretty messy.

Now x = 32º = 32π/180. So

Using the above approximation for sin x,

The accuracy is not guaranteed, since π/90 should be more places.

Actually, I couldn't bear this inaccuracy. So π/90 is .0349066 on my calculator, and if I hit the right buttons, sin
4
32º = .5299195 (and my calculator is OK). The remainder is (sin w)(.0349066) /4! We know sin w is less than
4
-8
1, so the remainder is less than (.0349066) /4! = 6.1861288 × 10 .
Even with these limited examples, we see different series get more accurate results with the same number of
terms. This can be studied in great detail.

Also, it is very convenient to know certain power series by heart. We will list the most important together with
region of convergence.