CK-12-Calculus

http://www.ck12.org Chapter 8. Infinite Series

Since 1.1 is close toπ 3 , we would try to find a Taylor Series of sinxatx 0 =π 3. Letf(x) =sinx. Then

f(x) =sinx,f′(x) =cosx,f′′(x) =−sinx,f′′′(x) =−cosxandf(π 3 )=

√ 3

2 ,f′

(π

3

)= 1

2 ,f′′

(π

3

)=−√ 3

2 ,f′′′

(π

3

)=

−^12.

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+ 1

Hence

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. 8912

We 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

1. Approximate ln 0.9 to 4 decimal places.


2. Approximate sin( 0. 8 )to 6 decimal places.
   Hint:consider centerπ 4


3. Approximate 913 to 6 decimal places.

Evaluating Non-Elementary 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.