Thomas' Calculus
Tenth Edition
Section 8.7- Taylor and Maclaurin Series
Lesson 33
1.
Find the Taylor polynomials of orders 0, 1, 2, and 3 for the function
     at    
The Taylor series for a function   at    is
  
  
     
   
    


      
          
            
          
The Taylor polynomial of order 0 is       
The Taylor polynomial of order 1 is           
          
The Taylor polynomial of order 2 is            
  

               





               



The Taylor polynomial of order 3 is


              




           


9.
Find the Maclaurin series for the function      
The Maclaurin series for the function      is


    







    

 


  
  
 




If we wanted to use summation notation,


 

since   
we can replace  by 
  
  

Therefore



 

 

     
    















13.
Find the Maclaurin series expansion for the function
          
  
       
     

 
 



   
               
            
             
          
     for   
The Maclaurin series expansion is


    

    
   
    







    

   
    




        .
Lesson 34- More Maclaurin Series
27.
Using the series in the Maclaurin series table as basic building blocks,
combine series expansions to find the Maclaurin series for

     

Obviously the Maclaurin series expansion for   is  
Similarly, the Maclaurin series expansion for   is   
The Maclaurin series expansion for   is














     is
The Maclaurin series expansion for








 













31.
Find the Maclaurin series expansion for   
Use the identity   
   
  
 








 
Now     

      




Therefore




 )

   

 
  




  
 



   

  


 








  
  
  
Or using summation notation we get

  
  
 


   
 
   


  





      
 the 1/2 cancels
 
  
with the 1st term in the summation.

and we obtain    
or
  



better yet, we can replace     by    to
get the simpler summation   




.
  