Some things to note:

Ageometric seriesis thesumof the numbers in a geometric progression:

We can find a simpler formula for this sum by multiplying both sides of the above equation by (1 −r), and we'll see that

since all the other terms cancel. Rearranging (for ) gives the convenient formula for a geometric series:

Note: If one were to begin the sum not from 0, but from a higher term, saym, then

now to do the problem:

a = 200, r = 0.4

let S14 be the sum of the first 14 terms

using

=> S14 = [200(1 - (0.4)^15)]/(1 - 0.4)

= 200(0.999998926)/0.6

= 333.3329753

since the sum to infinity is 333.3333333333333

the difference between the sum to infinity and the sum of the first 4 terms is:

333.3333333 - 333.3329753 = 0.000357967 = 3.58 x 10^-4 or 4 x 10^-4