The Chemistry Maths Book, Second Edition

7.6 MacLaurin and Taylor series 213

The nth derivative atx 1 = 10 isf

(n)

(0) 1 = 1 (−1)

n− 1

(n 1 − 1 1)!, and

The series converges when |x| 1 < 11. Thus,c

r

2 c

r+ 1

1 = 1 (r 1 + 1 1) 2 r 1 → 11 asr 1 → 1 ∞and by

the ratio test (7.18), the radius of convergence isR 1 = 11. In addition, the series is the

(convergent) alternating harmonic series whenx 1 = 11 :

4. 
   

   A list of some useful series

   
   


5. 
   

   |x| 1 < 11

   
   


6. 
   

   all x

   
   


7. 
   

   all x

   
   


8. 
   


9. 
   

   − 11 < 1 x 1 ≤ 1 + 1

   
   


10. 
    

    all x

    
    


11. 
    

    all x

    
    


12. 
    

    all x

    
    

0 Exercises 58–67

The Taylor series

12

The MacLaurin series, the expansion of a functionf(x) about the pointx 1 = 10 , is a

special case of the more general expansion of the function about the pointx 1 = 1 a:

coshx

xxx

=+

!

• 
  

!

• 
  

!

1 +

246

246



sinhxx

xxx

=+

!

• 
  

!

• 
  

!

• 
  

357

357



ex

xxx

x

=+ +

!

• 
  

!

• 
  

!

1 +

234

234



ln( ) 1

234

234

+=−+−+xx

xxx



−<<

ππ

22

x

tanxx

xx x

=+ + + +

35 7

3

2

15

17

315



cosx

xxx

=−

!

• 
  

!

−

!

1 +

246

246



sinxx

xxx

=−

!

• 
  

!

−

!

• 
  

357

357



()

() ()( )

11

1

2

12

3

23

+=++

−

!

• 
  

−−

!

xax +

aa

x

aa a

x

a



ln2 1

1

2

1

3

1

4

=− + − +

ln( ) 1

234

234

+=−+−+xx

xxx



12

Brook Taylor (1685–1731), secretary of the Royal Society. The series called after him appeared in his

Methodus incrementorum(1715), but had been known to James Gregory.