8.7. Taylor and Maclaurin Series http://www.ck12.org
This pattern repeats and limn→∞Rn(x) =0 can be checked as in the casex 0 =0. So the Taylor Series is
sinx=∞
n∑= 0 (−^1 )n√ 3
2 ( 2 n)!(
x−π 3) 2 n
+∞
n∑= 0 (−^1 )n
2 ( 2 n+ 1 )!(
x−π 3) 2 n+ 1Hence
sin( 1. 1 )≈∞
n∑= 0 (−^1 )n√ 3
2 ( 2 n)! (^0.^0528 )2 n+∑∞
n= 0(− 1 )n
2 ( 2 n+ 1 )!(^0.^0528 )2 n+ 1≈ 0. 86481823 + 0 .02638773 taking 2 terms from each sum
≈ 0. 8912We may also apply algebraic manipulation to standard Taylor Series.
Example 3Approximate 11. 92 to 4 decimal places.
Solution. There is standard Taylor Series:
( 1 −^1 x)^2 =∑∞k=^1 nxn−^1 for|x|<1 through term-by-term differentiation of the series for 1 −^1 x(( 1 −^1 x)^2 is inadequate at
x=− 0 .9).
Since 1.9 is close to 2, we consider
( 2 −^1 x)^2 =^14 ( 1 −^1 x
2 )^2 =(^14) ∑∞k= 1 n(x 2 )n−^1 for|x|<2.
So we takex= 0 .1 and then
1
1. 92 =
1
4
∞
n∑= 1 n(x
2)n− 1
≈^14 ( 1 + 2 ( 0. 05 )+ 3 ( 0. 05 )^2 + 4 ( 0. 05 )^3 )≈ 0. 2770.Exercises
- Approximate ln 0.9 to 4 decimal places.
- Approximate sin( 0. 8 )to 6 decimal places.
Hint:consider centerπ 4 - Approximate 913 to 6 decimal places.
Evaluating Nonelementary Integrals
There are many simple-looking functions that have no explicit formula for their integral in the form of elementary
functions. We could write their in-tegrals as Taylor Series in their interval of convergence.
Example 1Find a power series representation of∫exxdx
Solution. Sinceexxis not defined atx=0, we apply the Taylor Series ofexat, say,x=1 by writingex=e·ex−^1 with
a change of variableu=x−1.