Sequences and Series


   
 
Sigma Notation
The Greek letter S (read as sigma) denotes the sum. When written before the nth term of series, implies, the sum of all terms obtained by giving to n the different values 1,2,3…n. Thus,
 
 
 
 
 
 
Examples:
 
1. Find the sum to the series 1+2x+3x2+.... to n terms and to infinity when x < 1.
 
Suggested answer:
 
 
 
 
 
 
Subtracting (ii) from (i), we get
 
 
             
 
             
 
       
 
  
 
2. Find the sum to n terms of the series
 
 
Suggested answer:
 
 
nth term of (1,2,3,....) is n
 
 
 
 
 
Subtracting (ii) from (i), we get
 
 
 
         
 
 
3. Find the sum to infinity of the series
 
 
Suggested answer:
 
 
 
 
 
 
Subtracting (ii) from (i), we get
 
 
 
         
 
         
 
          
 
 
 
4. Sum the following:
 
i) 1.2 + 2.4 + 3.8 +.... to n terms
 
 
 
 
Suggested answer:
 
i) 1.2 + 2.4 + 3.8 +.... to n terms
 
The nth term of (1,2,3,....n) is n.
 
The nth term of (2,4,8,...) is
 
The nth term of the given series is n2n
 
 
     
 
 
 
Subtracting (ii) from (i), we have
 
 
               
 
               
 
               
 
 
  
 
 
 
 
Subtracting (ii) from (i), we have
 
 
 
         
 
 
 
Though it is not an arithmetic-geometric series, we can apply a similar method.
 
 
 
Subtracting (ii) from (i),
 
 
 
Subtracting (iv) from (iii),
 
 
 
 
 
 
 
Subtracting (ii) from (i),
 
 
 
 
 
find d.
 
Suggested answer:
 
 
 
Subtracting (ii) from (i), we get
 
 
 
 
 
 
Suggested answer:
 
 
 
 
 
  
     
   
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