Sum to Infinity

 

 

The sum to infinity is a finite value the sum of the first n terms of a geometric series tends to when n tends to infinite. Sum to infinity only exists when a series is convergent.

 

Sum to infinity is given by the expression:

                     

                and only exists if :

                                

 

Why?

          The sum of first n terms is given by:

                        

          As .    (only if )

          Hence .

          Thus the sum to infinity is given by

                                      

    If , , and the sum to infinity will not exist.

 

 

In a geometric progression, the common ratio is -1/3, and the sum of the first three terms is 14/27. Find

            (i)  the second negative term

            (ii) the sum to infinity

 

Solution:

 

      Sum of first three terms:

                        

                                  a = 2/3

 

      The second negative term is the fourth term:

                                    

 

      Sum to infinity:

                    

                     = 1/2

 

Cost of laying a cable comprises of a fixed amount of $5000 for every 50m and a variable cost of $3000 for the first 50m, $2400 for the next 50m, $1920 for the next 50m and so on.

        (i)  Calculate the total cost of laying a 300m long cable.

        (ii) Show that theoretically speaking the total variable cost cannot exceed

             $15000.

 

Solution:

 

      The variable cost is a G. P.

                 3000, 2400, 1920, ...

            with first term 3000 and common ratio 4/5.

 

      Total cost of laying a 300m cable

           = fixed cost + variable cost

           = 3000 x 5 + 3000 [1 - (4/5)⁶] 5

           = $ 41067.84

 

      Sum to infinity of variable cost

                                     

                          = 15000

Since the sum to infinity of the variable cost is $15000, the variable cost cannot exceed $15000.