http://www.ck12.org Chapter 8. Infinite Series
f′(x) =^12 ( 1 +x)−^12 ,f′′(x) =^12(
−^12
)
( 1 +x)−^32 =−^14 ( 1 +x)−^32 ,f′′′(x) =(
−^14
)(
−^32
)
( 1 +x)−^52 =^38 ( 1 +x)−^52 ,f(^4 )(x) =( 3
8
)(
−^52
)
( 1 +x)−^72 =^1516 ( 1 +x)−^72.For|x|≤ 0. 1 ,|f(^4 )(− 0. 1 )|=^1516 ( 1 +(−^10. 1 )) 72 ≤ 1 .5.
So|R 3 (x)|≤( 1. 5 )·4!^1 |(− 0. 1 )|^4 ≤^124.^5 ( 0. 1 )^4 ≈ 6. 25 × 10 −^6. This is the truncation error of approximating by the
third-degree Maclaurin polynomial.
The following applet illustrates approximating functions with Taylor Series. You can change the center of the series
and also observe how the error changes for the estimation at a particular value ofxwhere the error is f(x)−Tn(x).
Taylor Series and Polynomials Applet.
Exercises
- Find the power series representation off(x) =sinxatx=0 for allx. Why is it the Maclaurin series?
- Find the power series representation off(x) =cosxatx=π 3 for allx. Why is it the Taylor series atx=π 3?
- What is the truncation error of approximatingf(x) =
√
1 +xby its fourth-degree Maclaurin series in for
|x|≤ 0 .1.Combining Series, Eulers Formula
In many cases, we could find Taylor (Maclaurin) series of functions from existing series without going through the
proof that limn→∞Rn(x) =0 Examples are products, quotients and some sine and cosine functions.
Example 1Find the Maclaurin series off(x) =xsinxfor allx.
Solution. The Maclaurin series of sinxis sinx=∑∞n= 0 (−( 21 n)n+x 12 n)+!^1. So the Maclaurin series ofxsinxisxsinx=
x∑∞n= 0 (−( 21 n)n+x 12 )n+!^1 =∑∞n= 0 (−( 21 n)+nx 12 n)+!^2
Example 2Find the Maclaurin series off(x) =cos^2 xfor allx.
Solution. We could avoid multiplying the Maclaurin series of cosxwith itself, by applying: cos^2 x=^12 +^12 cos 2x
on the Maclaurin series of cos 2x=∑∞n= 0 (−^1 () 2 nn()^2 !x)^2 n, giving
cos^2 x=^12 +^12∞
n∑= 0(− 1 )n( 2 x)^2 n
( 2 n)! =^1 +∞
n∑= 0(− 1 )n( 2 x)^2 n
2 ·( 2 n)!For any real numberθ,eiθ=cos+isinθwherei=√−1 is the imaginary unit. This is theEuler’s Formula.
Euler’s formula combines the complementarysineandcosinefunctions into the simpler exponential function and
heavily applies the separation of real and imaginary parts of complex numbers.
Example 3Find the Maclaurin series of cosxand sinxfor allxthrougheix.
Solution. cosx+isinx=eixwhich has a Maclaurin series
eix=∞
n∑= 0(ix)n
n! =∞
m∑= 0(ix)^2 m
( 2 m)!+∞
m∑= 0(ix)^2 m+^1
( 2 m+ 1 )!